# GRE Quantitative - Integers

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This lesson is part of a series of lessons for the quantitative reasoning section of the GRE revised General Test. In this lesson, we will learn about

Integers
The integers are the numbers (1, 2, 3, … ), together with their negatives, (−1, −2, −3, …)  and zero, 0. Therefore, the set of integers consists of { … , −3, −2, −1, 0, 1, 2, 3, … }. The positive integers are greater than 0, the negative integers are less than 0, and 0 is neither positive nor negative. When integers are added, subtracted, or multiplied, the result is always an integer.

When adding two integers of the same sign

• Add their absolute values and
• Keep the sign of the original integers.

When adding two integers with different signs

• Find the absolute value of both.
• Subtract the smaller value from the larger value.
• The sign of the answer is the original sign of the number with the larger absolute value.

How to add integers using the number line and how to add integers using rules?

Subtracting Integers
We can rewrite a subtraction problem as an addition problem.
Subtracting a positive is the same as adding a negative.
xy = x + (−y)
Subtracting a negative is the same as adding a positive.
x – (−y)= x + y

How to write a subtraction problem as an addition problem and then use the rules for adding integers?

Multiplying integers
When multiplying two integers:
If the signs are the same, the product is positive.
(+)(+) = (+)
(−)(−) = (+)
If the signs are different, the product is negative.
(−)(+) = (−)
(+)(−) = (−)

When two or more integers are multiplied:

• The product of an even number of negative numbers is positive.
• The product of an odd number of negative numbers is negative.

Basic rules for multiplying integers and examples

Dividing integers
When dividing two integers:
If the signs are the same, the quotient is positive.
If the signs are the different, the quotient is negative.

Basic rules for dividing integers and examples

### Even and odd integers

Even integers are integers that are divisible by 2. Zero is considered an even integer. An even integer always ends in 0, 2, 4, 6, or 8. The set of even integers is {… , −12, −10, −8, −6, −4, −2, 0, 2, 4, 6, 8, 10, 12,  …}

Odd integers are integers that are not divisible by 2. An odd number always ends in 1, 3, 5, 7, or 9. Note that when a positive odd integer is divided by 2, the remainder is always 1. The set of odd integers is {… , −11, −9, −7, −5, −3, −1, 1, 3, 5, 7, 9, 11, …}

To tell whether a number is even or odd, look at the rightmost digit.
For example, the number 255 is an odd number because it ends in 5, which is odd. Likewise, 702 is an even number because it ends in 2, which is even.

There are several useful facts regarding the sum of even and odd integers.

• The sum of two even integers is an even integer.
• The sum of two odd integers is an even integer.
• The sum of an even integer and an odd integer is an odd integer.

There are several useful facts regarding the product of even and odd integers.

• The product of two even integers is an even integer.
• The product of two odd integers is an odd integer.
• The product of an even integer and an odd integer is an even integer.
 Addition examples: even + even = even 2 + 2 = 4 odd + odd = even 1 + 1 = 2 odd + even = odd 1 + 2 = 3 Multiplication examples: even × even = even 2 × 2 = 4 odd × odd = odd 1 × 1 = 1 odd × even = even 1 × 2 = 2

Examples of sum and product of even and odd integers and their patterns

### Factors and Multiples

In the problem, 3 × 10 = 30
3 and 10 are both factors or divisors of 30. 30 is a multiple of both 3 and 10.

A number is a factor of 30 if it can divide 30.
A list of all the positive factors of 30:  1, 2, 3, 5, 6, 10, 15, 30
A list of all the negative factors of 30:  −1, −2, −3, −5, −6, −10, −15, −30

A list of all positive multiples of 12 has no end: 0, 12, 24, 36, 48, etc. Similarly, every non-zero integer has infinitely many multiples.

• 1 is a factor of every integer; 1 is not a multiple of any integer except 1 and  −1.
• 0 is a multiple of every integer; 0 is not a factor of any integer except 0.

The difference between factors and multiples and how to find the factor and multiples
If you are looking for the factors, divide. If you are looking for the multiples, multiply.

### Least Common Multiple

The least common multiple (LCM) of two nonzero integers, a and b is the smallest positive integer that is a multiple of both a and b.

What is the LCM of 4 and 6?
Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, …
Multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, …
Common multiples of 4 and 6 are the numbers that are in both lists: 12, 24, 36, ….
So the least common multiple of 4 and 6 is the smallest one of those: 12

How to find the least common multiple of two or three numbers?
The LCM of more than two integers is the smallest number that is an integer multiple of each of them.

### Greatest Common Factor

The greatest common divisor or greatest common factor (GCF) of two nonzero integers, a and b is the greatest positive integer that is a divisor or factor of both a and b.

What is the GCF of 30 and 75?
Positive divisors or factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30
Positive divisors or factors of 75 are 1, 3, 5, 15, 25, and 75.
The common positive divisors or factors of 30 and 75 are 1, 3, 5, and 15
The greatest common factor is 15.

How to calculate factors of numbers and use those factors to find the GCF, Greatest Common Factor?

### Prime Numbers and Composite Numbers

A prime number is an integer greater than 1 that has only two positive divisors:  1 and itself. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

The integer 12 is not a prime number, since it has six positive divisors:  1, 2, 3, 4, 6, and 12. The integer 1 is not a prime number, and the integer 2 is the only prime number that is even.

An integer greater than 1 that is not a prime number is called a composite number. The first ten composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18.

How to distinguish prime numbers?

### Prime Factorization

Every integer greater than 1 either is a prime number or can be uniquely expressed as a product of factors that are prime numbers, or prime divisors. Such an expression is called a prime factorization. Here are several examples of prime factorizations.
12 = 2 × 2 × 3 = 22 × 3
14 = 2 × 7
51 = 3 × 17
81 = 3 × 3 × 3 × 3 = 34
338 = 2 × 13 × 13 = 2 × 132

How to get the prime factorization of a number?

Review of Integers for the GRE
This video is a review of basic arithmetic for the purpose of solving problems on the quantitative reasoning section of the GRE revised General Test. This video covers the basics of integers. Topics covered include: Number Lines, Operations, Absolute Values, Addition and Subtraction of integers.

Basics of integers
Topics covered include: Multiplication, Multiples, Factors (Divisors), Lowest Common Multiple (LCM), and the Greatest Common Divisor (GCD) also known as the Greatest Common Factor (GCF).

Basic Arithmetic for the GRE revised General Test
This video is a review of basic arithmetic for the purpose of solving problems on the quantitative reasoning section of the GRE revised General Test. This video covers the basics of integers. Topics covered include: Division, Division as a Quotient and Reminder and Divisibility of Integers.

This video is a review of basic arithmetic for the purpose of solving problems on the quantitative reasoning section of the GRE revised General Test. This video covers the basics of integers. Topics covered include: Even and Odd Integers, Sum of Even and Odd Integers, Product of Even and Odd Integers, Prime Numbers, Prime Factorization, and Composite Numbers.

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