problem solving on integer

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Math Problems and Solutions on Integers

Problems related to integer numbers in mathematics are presented along with their solutions.

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Integers Worksheets

Welcome to the integers worksheets page at Math-Drills.com where you may have a negative experience, but in the world of integers, that's a good thing! This page includes Integers worksheets for comparing and ordering integers, adding, subtracting, multiplying and dividing integers and order of operations with integers.

If you've ever spent time in Canada in January, you've most likely experienced a negative integer first hand. Banks like you to keep negative balances in your accounts, so they can charge you loads of interest. Deep sea divers spend all sorts of time in negative integer territory. There are many reasons why a knowledge of integers is helpful even if you are not going to pursue an accounting or deep sea diving career. One hugely important reason is that there are many high school mathematics topics that will rely on a strong knowledge of integers and the rules associated with them.

We've included a few hundred integers worksheets on this page to help support your students in their pursuit of knowledge. You may also want to get one of those giant integer number lines to post if you are a teacher, or print off a few of our integer number lines. You can also project them on your whiteboard or make an overhead transparency. For homeschoolers or those with only one or a few students, the paper versions should do. The other thing that we highly recommend are integer chips a.k.a. two-color counters. Read more about them below.

Most Popular Integers Worksheets this Week

General Use Printables

Four quadrant coordinate graph paper.

Coordinate graph paper can be very useful when studying integers. Coordinate geometry is a practical application of integers and can give students practice with using integers while learning another related skill. The coordinate graph paper in this section can also be found on the graph paper page, but it is included here for your convenience.

*More Coordinate Graph Paper Options on the Graph Paper Page*

Integer Number lines

Integer number lines can be used for various math activities including operations with integers, counting, comparing, ordering, etc.

Comparing & Ordering Integers Worksheets

Comparing and ordering integers worksheets for learning about ordinality in integers.

Comparing Random Pairs of Integers

For students who are just starting with integers, it is very helpful if they can use an integer number line to compare integers and to see how the placement of integers works. They should quickly realize that negative numbers are counter-intuitive because they are probably quite used to larger absolute values meaning larger numbers. The reverse is the case, of course, with negative numbers. Students should be able to recognize easily that a positive number is always greater than a negative number and that between two negative integers, the one with the lesser absolute value is actually the greater number. Have students practice with these integers worksheets and follow up with the close proximity comparing integers worksheets.

Comparing Pairs of Integers in Close Proximity

By close proximity, we mean that the integers being compared differ very little in value. Depending on the range, we have allowed various differences between the two integers being compared. In the first set where the range is -9 to 9, the difference between the two numbers is always 1. With the largest range, a difference of up to 5 is allowed. These worksheets will help students further hone their ability to visualize and conceptualize the idea of negative numbers and will serve as a foundation for all the other worksheets on this page.

Ordering and Sorting Integers

Arithmetic Operations with Integers

Adding integers worksheets.

Two-color counters are fantastic manipulatives for teaching and learning about integer addition. Two-color counters are usually plastic chips that come with yellow on one side and red on the other side. They might be available in other colors, so you'll have to substitute your own colors in the following description.

Adding with two-color counters is actually quite easy. You model the first number with a pile of chips flipped to the correct side and you also model the second number with a pile of chips flipped to the correct side; then you mash them all together, take out the zeros (if any) and behold, you have your answer! Need further elaboration? Read on!

The correct side means using red to model negative numbers and yellow to model positive numbers. You would model —5 with five red chips and 7 with seven yellow chips. Mashing them together should be straight forward although, you'll want to caution your students to be less exuberant than usual, so none of the chips get flipped. Taking out the zeros means removing as many pairs of yellow and red chips as you can. You can do this because —1 and 1 when added together equals zero (this is called the zero principle). If you remove the zeros, you don't affect the answer. The benefit of removing the zeros, however, is that you always end up with only one color and as a consequence, the answer to the integer question. If you have no chips left at the end, the answer is zero!

75 Question Worksheets

50 question worksheets, large print 25 question worksheets, vertically arranged integer addition.

Subtracting Integers Worksheets

Subtracting with integer chips is a little different. Integer subtraction can be thought of as removing. To subtract with integer chips, begin by modeling the first number (the minuend) with integer chips. Next, remove the chips that would represent the second number from your pile and you will have your answer. Unfortunately, that isn't all there is to it. This works beautifully if you have enough of the right color chip to remove, but often times you don't. For example, 5 - (-5), would require five yellow chips to start and would also require the removal of five red chips, but there aren't any red chips! Thank goodness, we have the zero principle. Adding or subtracting zero (a red chip and a yellow chip) has no effect on the original number, so we could add as many zeros as we wanted to the pile, and the number would still be the same. All that is needed then is to add as many zeros (pairs of red and yellow chips) as needed until there are enough of the correct color chip to remove. In our example 5 - (-5), you would add 5 zeros, so that you could remove five red chips. You would then be left with 10 yellow chips (or +10) which is the answer to the question.

Vertically Arranged Integer Subtraction

Mixed Adding and Subtracting Integers Worksheets

The worksheets in this section include addition and subtraction on the same page. Students will have to pay close attention to the signs and apply their knowledge of integer addition and subtraction to each question. The use of counters or number lines could be helpful to some students.

Scaffolded Integer Addition and Subtraction

These worksheets include groups of questions that all result in positive or negative sums or differences. They can be used to help students see more clearly how certain integer questions end up with positive and negative results. In the case of addition of negative and positive integers, some people suggest looking for the "heavier" value to determine whether the sum will be positive of negative. More technically, it would be the integer with the greater absolute value. For example, in the question (−2) + 5, the absolute value of the positive integer is greater, so the sum will be positive.

In subtraction questions, the focus is on the subtrahend (the value being subtracted). In positive minus positive questions, if the subtrahend is greater than the minuend, the answer will be negative. In negative minus negative questions, if the subtrahend has a greater absolute value, the answer will be positive. Vice-versa for both situations. Alternatively, students can always convert subtraction questions to addition questions by changing the signs (e.g. (−5) − (−7) is the same as (−5) + 7; 3 − 5 is the same as 3 + (−5)).

Multiplying Integers Worksheets

Multiplying integers is very similar to multiplication facts except students need to learn the rules for the negative and positive signs. In short, they are:

In words, multiplying two positives or two negatives together results in a positive product, and multiplying a negative and a positive in either order results in a negative product. So, -8 × 8, 8 × (-8), -8 × (-8) and 8 × 8 all result in an absolute value of 64, but in two cases, the answer is positive (64) and in two cases the answer is negative (-64).

Should you wish to develop some "real-world" examples of integer multiplication, it might be a stretch due to the abstract nature of negative numbers. Sure, you could come up with some scenario about owing a debt and removing the debt in previous months, but this may only result in confusion. For now students can learn the rules of multiplying integers and worry about the analogies later!

100 Question Worksheets

Dividing integers worksheets.

Luckily (for your students), the rules of dividing integers are the same as the rules for multiplying:

Dividing a positive by a positive integer or a negative by a negative integer will result in a positive integer. Dividing a negative by a positive integer or a positive by a negative integer will result in a negative integer. A good grasp of division facts and a knowledge of the rules for multiplying and dividing integers will go a long way in helping your students master integer division. Use the worksheets in this section to guide students along.

Mixed Multiplying and Dividing Integers Worksheets

This section includes worksheets with both multiplying and dividing integers on the same page. As long as students know their facts and the integer rules for multiplying and dividing, their sole worry will be to pay attention to the operation signs.

All Operations with Integers Math Worksheets

In this section, the integers math worksheets include all of the operations. Students will need to pay attention to the operations and the signs and use mental math or another strategy to arrive at the correct answers. It should go without saying that students need to know their basic addition, subtraction, multiplication and division facts and rules regarding operations with integers before they should complete any of these worksheets independently. Of course, the worksheets can be used as a source of questions for lessons, tests or other learning activities.

Order of Operations with Integers Worksheets

The order of operations worksheets in this section actually reside on the Order of Operations page, but they are included here for your convenience.

Copyright © 2005-2023 Math-Drills.com You may use the math worksheets on this website according to our Terms of Use to help students learn math.

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How to Solve Integers and Their Properties

Last Updated: June 30, 2018

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, volunteer authors worked to edit and improve it over time. This article has been viewed 27,153 times. Learn more...

An integer is a set of natural numbers, their negatives, and zero. However, some integers are natural numbers, including 1, 2, 3, and so on. Their negative values are, -1, -2, -3, and so on. So integers are the set of numbers including (…-3, -2, -1, 0, 1, 2, 3,…). An integer is never a fraction, decimal, or percentage, it can only be a whole number. To solve integers and use their properties, learn to use addition and subtraction properties and use multiplication properties.

Using Addition and Subtraction Properties

• a + b = c (where both a and b are positive numbers the sum c is also positive)
• For example: 2 + 2 = 4

• -a + -b = -c (where both a and b are negative, you get the absolute value of the numbers then you proceed to add, and use the negative sign for the sum)
• For example: -2+ (-2)=-4

• a + (-b) = c (when your terms are of different signs, determine the larger number's value, then get the absolute value of both terms and subtract the lesser value from the larger value. Use the sign of the larger number for the answer.)
• For example: 5 + (-1) = 4

• -a +b = c (get the absolute value of the numbers and again, proceed to subtract the lesser value from the larger value and assume the sign of the larger value)
• For example: -5 + 2 = -3

• An example of the additive identity is: a + 0 = a
• Mathematically, the additive identity looks like: 2 + 0 = 2 or 6 + 0 = 6

• The additive inverse is when a number is added to the negative equivalent of itself.
• For example: a + (-b) = 0, where b is equal to a
• Mathematically, the additive inverse looks like: 5 + -5 = 0

• For example: (5+3) +1 = 9 has the same sum as 5+ (3+1) = 9

Using Multiplication Properties

• When a and b are positive numbers and not equal to zero: +a * + b = +c
• When a and b are both negative numbers and not equal to zero: -a*-b = +c

• However, understand that any number multiplied by zero, equals zero.

• For example: a(b+c) = ab + ac
• Mathematically, this looks like: 5(2+3) = 5(2) + 5(3)
• Note that there is no inverse property for multiplication because the inverse of a whole number is a fraction, and fractions are not an element of integer.

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Integer Word Problems Worksheets

An integer is defined as a number that can be written without a fractional component. For example, 11, 8, 0, and −1908 are integers whereas √5, Π are not integers. The set of integers consists of zero, the positive natural numbers, and their additive inverses. Integers are closed under the operations of addition and multiplication . Integer word problems worksheets provide a variety of word problems associated with the use and properties of integers. 

Benefits of Integers Word Problems Worksheets

We use integers in our day-to-day life like measuring temperature, sea level, and speed limit. Translating verbal descriptions into expressions is an essential initial step in solving word problems. Deposits are normally represented by a positive sign and withdrawals are denoted by a negative sign. Negative numbers are used in weather forecasting to show the temperature of a region. Solving these integers word problems will help us relate the concept with practical applications.

Download Integers Word Problems Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

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• 6th Grade Integers Worksheets
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In Mathematics, integers are the collection of whole numbers and negative numbers. Similar to whole numbers, integers also does not include the fractional part. Thus, we can say, integers are numbers that can be positive, negative or zero, but cannot be a fraction . We can  perform all the  arithmetic operations , like addition, subtraction, multiplication and division, on integers. The examples of integers are, 1, 2, 5,8, -9, -12, etc. The symbol of integers is “ Z “. Now, let us discuss the definition of integers, symbol, types, operations on integers, rules and properties associated to integers, how to represent integers on number line with many solved examples in detail.

What are Integers?

The word integer originated from the Latin word “Integer” which means whole or intact. Integers is a special set of numbers comprising zero, positive numbers and negative numbers.

Examples of Integers: – 1, -12, 6, 15.

The integers are represented by the symbol ‘ Z’. 

Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}

Types of Integers

Integers come in three types:

• Positive Integers (Natural numbers)
• Negative Integers (Additive inverse of Natural Numbers)

Zero is neither a positive nor a negative integer. It is a neutral number i.e. zero has no sign (+ or -).

Positive Integers

The positive integers are the natural numbers or also called counting numbers. These integers are also sometimes denoted by Z + . The positive integers lie on the right side of 0 on a number line.

Negative Integers

The negative integers are the negative of natural numbers. They are denoted by Z – . The negative integers lie on the left side of 0 on a number line.

How to Represent Integers on Number Line?

As we have already discussed the three categories of integers, we can easily represent them on a number line based on positive integers, negative integers and zero. 

Zero is the centre of integers on a number line. Positive integers lie on the right side of zero and negative integers lie on the left. See the below figure.

Rules of Integers

Rules defined for integers are:

• Sum of two positive integers is an integer
• Sum of two negative integers is an integer
• Product of two positive integers is an integer
• Product of two negative integers is an integer
• Sum of an integer and its inverse is equal to zero
• Product of an integer and its reciprocal is equal to 1

Arithmetic Operations on Integers

The basic Maths operations performed on integers are:

• Addition of integers
• Subtraction of integers
• Multiplication of integers
• Division of integers

Addition of Integers

While adding the two integers with the same sign, add the absolute values, and write down the sum with the sign provided with the numbers.

For example,

(+4) + (+7) = +11

(-6) + (-4) = -10

While adding two integers with different signs, subtract the absolute values, and write down the difference with the sign of the number which has the largest absolute value.

(-4) + (+2) = -2

(+6) + (-4) = +2.

Subtraction of Integers

While subtracting two integers, change the sign of the second number which is being subtracted, and follow the rules of addition.

(-7) – (+4) = (-7) + (-4) = -11

(+8) – (+3) = (+8) + (-3) = +5

Multiplication of Integers

While multiplying two integer numbers, the rule is simple.

• If both the integers have the same sign, then the result is positive.
• If the integers have different signs, then the result is negative.
• (+2) x (+3) = +6
• (+3) x (-4) = – 12

Thus, we can summarise the multiplication of two integers with examples in the below table.

Division of Integers

The rule for dividing integers is similar to multiplication.

• (+6) ÷ (+2) = +3
• (-16) ÷ (+4) = -4

Properties of Integers

The major Properties of Integers are:

Closure Property

Associative property, commutative property.

• Distributive Property

Additive Inverse Property

• Multiplicative Inverse Property
• Identity Property

According to the closure property of integers, when two integers are added or multiplied together, it results in an integer only. If a and b are integers, then:

• a + b = integer
• a x b = integer

  Examples:  

• 2 + 5 = 7 (is an integer)
• 2 x 5 = 10 (is an integer)

According to the commutative property of integers, if a and b are two integers, then:

• a + b = b + a
• a x b = b x a
• 3 + 8 = 8 + 3 = 11
• 3 x 8 = 8 x 3 = 24

But for the commutative property is not applicable to subtraction and division of integers. 

As per the associative property , if a, b and c are integers, then:

• a+(b+c) = (a+b)+c
• ax(bxc) = (axb)xc
• 2+(3+4) = (2+3)+4 = 9
• 2x(3×4) = (2×3)x4 = 24

Similar to commutativity, associativity is applicable for the addition and multiplication of integers only.

Distributive property

According to the distributive property of integers, if a, b and c are integers, then:

a x (b + c) = a x b + a x c

Example: Prove that: 3 x (5 + 1) = 3 x 5 + 3 x 1

LHS = 3 x (5 + 1) = 3 x 6 = 18

RHS = 3 x 5 + 3 x 1 = 15 + 3 = 18

Since, LHS = RHS

Hence, proved.

If a is an integer, then as per the additive inverse property of integers, 

a + (-a) = 0

Hence, -a is the additive inverse of integer a.

Multiplicative inverse Property

If a is an integer, then as per the multiplicative inverse property of integers, 

a x (1/a) = 1

Hence, 1/a is the multiplicative inverse of integer a.

Identity Property of Integers

The identity elements of integers are:

Example: -100,-12,-1, 0, 2, 1000, 989 etc…

As a set, it can be represented as follows:

Applications of Integers

Integers are not just numbers on paper; they have many real-life applications. The effect of positive and negative numbers in the real world is different. They are mainly used to symbolize two contradicting situations.

For example, when the temperature is above zero, positive numbers are used to denote temperature, whereas negative numbers indicate the temperature below zero. They help one to compare and measure two things like how big or small or more or fewer things are and hence can quantify things.

Some real-life situations where integers come into play are player’s scores in golf, football and hockey tournaments, the rating of movies or songs, in banks credits and debits are represented as positive and negative amounts respectively.

Integers Related Articles

• Properties Of Integers
• Addition and Subtraction of Integers
• Multiplication and Division of Integers
• Consecutive Integers
• Integers As Exponents
• Operations Of Integers
• Word Problems On Integers

Examples on Integers

Plot the following integers on the number line:

-121, -97, -82, -67, -43, -10, 0, 10, 36, 55, 64, 77, 110, 126, 147.

In the above number line, each number has been plotted with a red dot.

Solve the following:

• 5 + (-3) = ?
• (-5) + (-3) = ?
• (-5) x (-3) = ?
• 5 + (-3) = 5 – 3 = 2
• (-5) + (-3) = -5 – 3 = -8
• (-5) x (-3) = 15

Solve the following product of integers:

• (+5) × (+10)
• (- 5) × (7)
• (+5) × (+10) = +50
• (12) × (5) = 60
• (- 5) × (7) = -35
• 5 × (-4) = -20

Solve the following division of integers:

• (-9) ÷ (-3)
• (-18) ÷ (3)
• (4000) ÷ (- 100)
• (-9) ÷ (-3) = 3
• (-18) ÷ (3) = -6
• (4000) ÷ (- 100) = -40

Practice Questions on Integers

1. Sum of two positive integers is a positive integer. True or False? 2. What is the sum of first five positive integers? 3. What is the product of first five positive odd integers? 4. Plot the integers from -10 to +10 on the number line.

Frequently Asked Questions on Integers

What are integers, what is an integer formula, what are the examples of integers, can integers be negative, what are the types of integers.

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The set of positive and negative integers together with 0 is called Please answer. Please help me byjus.

The set of positive natural numbers and negative numbers together with 0 is called Integers.

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Module 10: Linear Equations

Using a problem-solving strategy to solve number problems, learning outcomes.

• Solve number problems
• Solve consecutive integer problems

Solving Number Problems

Now we will translate and solve number problems. In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the Problem-Solving Strategy. Remember to look for clue words such as difference , of , and and .

The difference of a number and six is thirteen. Find the number.

https://ohm.lumenlearning.com/multiembedq.php?id=142763&theme=oea&iframe_resize_id=mom50

The sum of twice a number and seven is fifteen. Find the number.

Show Solution

https://ohm.lumenlearning.com/multiembedq.php?id=142770&theme=oea&iframe_resize_id=mom60

Watch the following video to see another example of how to solve a number problem.

Solving for Two or More Numbers

Some number word problems ask you to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. We will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

One number is five more than another. The sum of the numbers is twenty-one. Find the numbers.

https://ohm.lumenlearning.com/multiembedq.php?id=142775&theme=oea&iframe_resize_id=mom70

Watch the following video to see another example of how to find two numbers given the relationship between the two.

The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

https://ohm.lumenlearning.com/multiembedq.php?id=142806&theme=oea&iframe_resize_id=mom80

One number is ten more than twice another. Their sum is one. Find the numbers.

https://ohm.lumenlearning.com/multiembedq.php?id=142811&theme=oea&iframe_resize_id=mom90

Solving for Consecutive Integers

Another type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other.  Some examples of consecutive integers are:

[latex]\begin{array}{c} \hfill \text{…}1, 2, 3, 4, 5, 6\text{,…}\hfill \end{array}[/latex] [latex]\text{…}-10,-9,-8,-7\text{,…}[/latex] [latex]\text{…}150,151,152,153\text{,…}[/latex]

If we are looking for several consecutive numbers, it is important to first identify what they look like with variables before we set up the equation.  Notice that each number is one more than the number preceding it. So if we define the first integer as [latex]n[/latex], the next consecutive integer is [latex]n+1[/latex]. The one after that is one more than [latex]n+1[/latex], so it is [latex]n+1+1[/latex], or [latex]n+2[/latex].

[latex]\begin{array}{cccc}n\hfill & & & \text{1st integer}\hfill \\ n+1\hfill & & & \text{2nd consecutive integer}\hfill \\ n+2\hfill & & & \text{3rd consecutive integer}\hfill \end{array}[/latex]

For example, let’s say I want to know the next consecutive integer after [latex]4[/latex]. In mathematical terms, we would add [latex]1[/latex] to [latex]4[/latex] to get [latex]5[/latex]. We can generalize this idea as follows: the consecutive integer of any number, [latex]x[/latex], is [latex]x+1[/latex]. If we continue this pattern, we can define any number of consecutive integers from any starting point. The following table shows how to describe four consecutive integers using algebraic notation.

We apply the idea of consecutive integers to solving a word problem in the following example.

The sum of two consecutive integers is [latex]47[/latex]. Find the numbers.

The sum of three consecutive integers is [latex]93[/latex]. What are the integers?

• Read and understand:  We are looking for three numbers, and we know they are consecutive integers.
• Constants and Variables: [latex]93[/latex] is a constant. The first integer we will call [latex]x[/latex]. Second integer: [latex]x+1[/latex] Third integer: [latex]x+2[/latex]
• Translate:  The sum of three consecutive integers translates to [latex]x+\left(x+1\right)+\left(x+2\right)[/latex], based on how we defined the first, second, and third integers. Notice how we placed parentheses around the second and third integers. This is just to make each integer more distinct. “ is 93 ” translates to “[latex]=93[/latex]” since “ is ” is associated with equals.
• Write an equation:  [latex]x+\left(x+1\right)+\left(x+2\right)=93[/latex]

[latex]x+x+1+x+2=93[/latex]

Combine like terms, simplify, and solve.

[latex]\begin{array}{r}x+x+1+x+2=93\\3x+3 = 93\\\underline{-3\,\,\,\,\,-3}\\3x=90\\\frac{3x}{3}=\frac{90}{3}\\x=30\end{array}[/latex]

• Check and Interpret: Okay, we have found a value for [latex]x[/latex]. We were asked to find the value of three consecutive integers, so we need to do a couple more steps. Remember how we defined our variables:

The first integer we will call [latex]x[/latex], [latex]x=30[/latex] Second integer: [latex]x+1[/latex] so [latex]30+1=31[/latex] Third integer: [latex]x+2[/latex] so [latex]30+2=32[/latex] The three consecutive integers whose sum is [latex]93[/latex] are [latex]30\text{, }31\text{, and }32[/latex]

Find three consecutive integers whose sum is [latex]42[/latex].

Watch this video for another example of how to find three consecutive integers given their sum.

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Solve the following integer programming problem using branch and...

Solve the following integer programming problem using branch and bound method. Use table/graphs in a sheet of paper when answering.

Answer & Explanation

Branch and bound method for integer programming problem:

Solve the linear programming relaxation of the problem. This will give you an upper bound on the optimal integer solution.

Branch on one of the fractional variables in the relaxed solution. This will create two new subproblems, one where the variable is rounded down and one where it is rounded up.

Solve each of the subproblems recursively. If a subproblem is infeasible, discard it. Otherwise, update the upper bound if necessary.

Repeat steps 2 and 3 until all subproblems have been solved. The best solution found so far is the optimal solution to the original problem.

The branch and bound method is a systematic way to solve integer programming problems by iteratively partitioning the feasible solution space into smaller subsets and solving the relaxed linear programming problems associated with each subset.

Solution to the integer programming problem using branch and bound method:

Step 1: Create a table of feasible solutions.

The feasible solutions are all the points that satisfy all of the constraints. We can use the following table to keep track of the feasible solutions:

Step 2: Branch and bound.

We start by branching on the first constraint,  6 x 1 + 7 x 2 < = 4 2 . We create two new nodes, one for  x 1 < = 7 and one for  x 1 > 7 .

Node 1:  x 1 < = 7

We can now solve the linear programming problem with the constraint  x 1 < = 7 . This gives us the following feasible solutions:

Node 2:  x 1 > 7

We can now solve the linear programming problem with the constraint  x 1 > 7 . This gives us the following feasible solutions:

Step 3: Compare the feasible solutions from each node.

The best feasible solution from node 1 is  x 1 = 6 and x2 = 0, with a value of z = 0 . The best feasible solution from node 2 i s x 1 = 1 0 and x 2 = 1 , with a value of  z = 3 0 .

Step 4: Return the best feasible solution.

The best feasible solution is  x 1 = 1 0 and  x 2 = 1 , with a value of  z = 3 0 .

Therefore, the optimal solution to the integer programming problem is x 1 = 1 0 and x 2 = 1 , with a value of z = 3 0 .

Key references

Baaijens, J. A., Bonizzoni, P., Boucher, C., Della Vedova, G., Pirola, Y., Rizzi, R., & Sirén, J. (2022). Computational graph pangenomics: a tutorial on data structures and their applications. Natural computing , 21 (1), 81-108.

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Algebra Word Problems: Integers

Things to watch out for: be careful when translating the sentences into equations.

For example: “John has 5 fewer sweets than twice the number that Alice has”         is translated as j = 2 a –5         and not j = 5 – 2 a

Although, the number 5 is mentioned first in the sentence that does not mean that it would come first in the equation. Read the sentence carefully. In these lessons, we will cover some examples of integer word problems with two unknowns. Refer to the following related topics for other types of integer word problems.

Integer Problems With Two Unknowns

A team won 3 times as many matches as it lost. If it won 15 matches, how many games did it lose?

Step 1: Sentence: A team won 3 times as many matches as it lost.

Assign variables :

Sentence: It won 15 matches 3x = 15

Step 2: Solve the equation

Initially, there were the same number of blue marbles and red marbles in a bag. John took out 5 blue marbles and then there were twice as many red marbles as blue marbles in the bag. How many red marbles are there in the bag?

Step 1: Assign variables :

Let x = red marbles

Sentence: Initially, blue marbles = red marbles = x , then John took out 5 blue marbles. x – 5 = blue marbles

Sentence: twice as many red marbles as blue marbles in the bag x = 2( x –5)

x = 2( x –5)

Remove the brackets x = 2 x – 10

Isolate variable x x = 10

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Mathematics > Optimization and Control

Title: differentiable cutting-plane layers for mixed-integer linear optimization.

Abstract: We consider the problem of solving a family of parametric mixed-integer linear optimization problems where some entries in the input data change. We introduce the concept of cutting-plane layer (CPL), i.e., a differentiable cutting-plane generator mapping the problem data and previous iterates to cutting planes. We propose a CPL implementation to generate split cuts, and by combining several CPLs, we devise a differentiable cutting-plane algorithm that exploits the repeated nature of parametric instances. In an offline phase, we train our algorithm by updating the internal parameters controlling the CPLs, thus altering cut generation. Once trained, our algorithm computes, with predictable execution times and a fixed number of cuts, solutions with low integrality gaps. Preliminary computational tests show that our algorithm generalizes on unseen instances and captures underlying parametric structures.

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Integer Calculator

What is an integer in math – integer definition, negative and positive number line, adding and subtracting integers, multiplying integers and integer division, exponents, roots, and logarithms, using the integer calculator.

Welcome to Omni's integer calculator , where we'll learn all about the four basic arithmetic operations: adding and subtracting integers , multiplying integers , and integer division . Then, we'll move to more difficult operations such as exponents , roots , and logarithms . Remember that the integer definition allows more than just positive integers, so this tool serves as a negative number calculator as well.

So what is an integer in math? What's a negative number squared or the log of a negative number?

The answer to that and so much more are right below !

🙋 Want to learn how to handle complex mathematical problems that involve more than one arithmetic operation? Check our distributive property calculator .

Colloquially speaking, an integer is a whole number . In other words, they don't include fractions (simple or decimal: if you're not sure what either of these terms means, go quickly to our decimal to fraction calculator ) or irrational numbers such as the π from circle calculations. The following are all examples of integers:

2 , 2021 , 13 , -17 , -173,029 , 0 , 1,000,000,000 .

• All positive integers, however long, are integers: we sometimes call them natural numbers ;
• Zero is also an integer ; and
• Integers include negative numbers as long as they don't contain fractions.

The formal integer definition is the following:

💡 An integer is any real number that we can write without a fractional part.

Let's stress the word " can " in the above integer definition. Observe how the simple fraction 4/2 is equal to 2 , so it is an integer by fraction simplification, although it doesn't look like it at first glance.

Before we move on to the operations, let's spare a separate section to the differences between positive integers and negative ones .

Integers (and all other real numbers) appear on an infinite axis called the number line .

In essence, the line tells us where one number lies with respect to the others : is it larger (to the right) or smaller (to the left) of something else? When they introduce us to mathematics, we count to ten on our fingers, so we know that, for example, 2 comes after 1 but before 3 .

Negative numbers are the mirror image of positive ones with the mirror put at 0 . In other words, if we start at zero and go right, we'll visit 1 , then 2 , 3 , and so on. On the other hand, if we go left, we meet the same numbers but with minuses: -1 , then -2 , -3 , and so on. This way, a number and its opposite are at the same distance from 0 but to the opposite sides (this distance is called the number's absolute value).

Arithmetic and algebraic properties apply to all the values on the negative and positive number line. In particular, we can add, subtract, multiply, divide, raise to a power, take the root, calculate the logarithm, etc., using those numbers. The differences in negative and positive number rules are small, and we point them out in each section below.

Given a positive integer, we can also find the sum of digits in order to determine the divisibility of the number.

When adding and subtracting integers, it's a good idea to keep the negative and positive number line from the above section in mind.

Suppose that we have integers a and b , and let's explain how we can find a + b and a - b .

• Look for a on the negative and positive number line.
• To the right , if b is positive; or
• To the left , if b is negative.
• To the left , if b is positive; or
• To the right , if b is negative.

See a few examples of adding and subtracting integers below:

10 + (-12) = 10 - 12 = -2

-3 + (-7) = -3 - 7 = -10

3 - (-2) = 3 + 2 = 5

-10 - (-4) = -10 + 4 = -6

Observe how whenever we had two signs next to each other, we have to put the negative number in brackets . Furthermore, it's possible to reduce the two into one in such case according to the following rules:

• + and + gives + ;
• + and - gives - ;
• - and + gives - ; and
• - and - gives + .

The first one doesn't really happen here, but it'll come in handy in the next section .

In essence, the negative and positive number rules for multiplying integers and integer division are almost the same . The only thing we have to keep in mind is the sign . To be precise, the result's sign depends on those of the factors or of the dividend and divisor for multiplication and division, respectively.

On the other hand, the result's value itself, be it positive or negative, doesn't care much about the signs . As such, we can begin our calculations as if both integers were positive, compute what the result would be in that case, and only then fix the sign accordingly. And by " accordingly ," we mean the same negative and positive number rules from the above section .

Below, we give a few examples of multiplying integers, followed by some integer division.

-4 × 5 = -20

10 × (-2) = -20

(-1) × (-8) = 8

24 / (-8) = -3

-7 / (-2) = 3.5

That concludes the four basic operations covered by Omni's integer calculator (or negative number calculator, if you prefer). Now, we move on to more complicated (but still simple!) algebraic expressions.

For positive integer exponents, the negative, and positive number rules are the same: the result is simply the number multiplied several times . And we've already seen how multiplying integers works in the above section , so let's simply mention a few examples:

4 3 = 4 × 4 × 4 = 64

(-3) 4 = (-3) × (-3) × (-3) × (-3) = 81

(-2) 5 = (-2) × (-2) × (-2) × (-2) × (-2) = -32

Observe how, for negative exponent bases, the result's sign depends on the parity of the power . It's a straightforward consequence of the negative and positive number rules from the adding and subtracting integers section . In particular, a negative number squared will always give a positive value.

Now, if the exponent is negative, we first get rid of its minus sign by changing the base into its multiplicative inverse : a -b = (1/a) b

From there, we repeat the usual thing while remembering the rules for multiplying fractions. For example:

4 -3 = (¼) 3 = ¼ × ¼ × ¼ = 1 / 64

(-3) -4 = (-⅓) 4 = (-⅓) × (-⅓) × (-⅓) × (-⅓) = 1 / 81

(-2) -5 = (-½) 5 = (-½) × (-½) × (-½) × (-½) × (-½) = - 1 / 32

Taking the root (also called the radical) is the opposite operation to the exponent . As such, some rules apply to both. Most importantly, note how for exponents, even powers always give a positive result , no matter the base's sign. If we translate that property to roots, we'll get that radicals of even order exist only for positive numbers . In particular, you can't have the square root of a negative number. In fact, such things do exist, but they're no longer real numbers but complex numbers.

Let's look at a couple of examples of integer roots:

∛(-125) = -5

Here the case is very simple: logarithms are defined only for positive numbers . In other words, there is no such thing as the log of a negative number . Again, similarly to the roots, in fact, there is, but it goes beyond real numbers, and the story gets quite difficult. If you are curious, check out our complex number calculator

Let's finish this part of the section with a couple of log examples with positive integers:

log 10 1000 = 3

log 9 6561 = 4

log 2 128 = 7

For completeness, let's finish with a quick instruction on how to use the integer calculator (or the negative number calculator, if you prefer).

To make your lives easier, we've prepared a nice step-by-step instruction on how to use Omni's integer calculator.

At the top of our tool, choose the operation you'd like to perform. There are seven options:

• Subtraction ;
• Multiplication ;
• Logarithm .

Once you select the operation, a symbolic formula appears underneath with a and b as variables.

Following the formula, input the values of a and b in the corresponding fields.

Read off the result from underneath.

For the four arithmetic operations: If you'd like to see the calculations described step by step, visit the appropriate Omni tool from the list below the result.

You might call it, " Integer calculator instructions ," but we call it, " Five easy steps to happiness and quick maths ." Whatever the name, we hope the tool saves you some time and helps with daily homework.

Is 0 an integer?

Yes. By definition, integers consist of all positive whole numbers (i.e., 1, 2, 3, and so on), their opposites (i.e., -1, -2, -3, and so on), and zero. The first group are the positive integers, the second are the negative ones, while 0 is neither positive nor negative.

Are all integers rational numbers?

Yes. By definition, we can represent all rational numbers as a simple fraction with integers in the numerator and denominator. As such, an integer a is equal to a / 1 , which surely satisfies the condition.

Obviously, it doesn't work the other way: not all rational numbers are integers , e.g., ½.

Can I take the square root of a negative number?

No , unless you're working with complex numbers. Both a positive number and a negative number squared give positive values, so there's no way to get the square root of a negative number (or any other even-ordered root, for that matter).

Can a negative number be an integer?

Yes. By definition, integers consist of all positive whole numbers (i.e., 1, 2, 3, and so on), their opposites (i.e., -1, -2, -3, and so on), and zero. The second group clearly involves negative numbers.

Can a fraction be an integer?

Yes. By definition, an integer is a number that we can express without fractional expressions. For instance, the number 2 is clearly an integer, but we can also write it as 4 / 2 , which is a fraction but a reducible one.

How do I add integers?

To add integers a and b , you need to:

• The point you end up in is the sum .
• Enjoy your result of adding integers.

How do I subtract integers?

To subtract integers a and b , you need to:

• The point you end up in is the difference .
• Enjoy your result of subtracting integers.

Are all integers natural numbers?

No. By definition, integers consist of all positive whole numbers (i.e., 1, 2, 3, and so on), their opposites (i.e., -1, -2, -3, and so on), and zero. Out of those, natural numbers are only the first set. Some people also include zero as a natural number, although not all scientists agree.

How do I multiply integers?

To multiply integers a and b , you need to:

• Multiply a and b as if they didn't have any signs.
• If both factors were positive or both negative, the result is positive ; and
• If one factor was positive and one negative, the result is negative .
• Enjoy your result of multiplying integers.

How do I divide integers?

For integer division of a and b , you need to:

• Divide a and b as if they didn't have any signs.
• If both numbers were positive or both negative, the result is positive ; and
• If one number was positive and one negative, the result is negative .
• Enjoy your result of integer division.

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