## Topic: Integers – Number Properties (Part 1)

### Introduction to Integers

Integers are whole numbers that include both positive and negative numbers, as well as zero. The set of integers can be represented as {…, -3, -2, -1, 0, 1, 2, 3, …}. Integers are the foundation of many mathematical concepts and play a significant role in the GMAT quantitative section.

### Basic Number Properties of Integers

There are several basic number properties of integers that are essential to understand for the GMAT:

a. Closure Property: Any two integers’ sum, difference, and product will always result in an integer. However, the division of two integers does not necessarily result in an integer.

b. Commutative Property: For any integers a and b, a + b = b + a, and a × b = b × a.

c. Associative Property: For any integers a, b, and c, (a + b) + c = a + (b + c), and (a × b) × c = a × (b × c).

d. Distributive Property: For any integers a, b, and c, a × (b + c) = a × b + a × c.

e. Identity Property: For any integer a, a + 0 = a and a × 1 = a.

f. Inverse Property: For any integer a, a + (-a) = 0 and a × (1/a) = 1, provided a ≠ 0.

### Divisibility Rules

Divisibility rules help in determining whether a number is divisible by another number without performing long division. The most common divisibility rules tested on the GMAT include:

a. Divisible by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).

b. Divisible by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.

c. Divisible by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

d. Divisible by 5: A number is divisible by 5 if its last digit is either 0 or 5.

e. Divisible by 6: A number is divisible by 6 if it is divisible by both 2 and 3.

f. Divisible by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.

### Factors and Multiples

a. Factors: A factor of an integer n is an integer that divides n without leaving a remainder. For example, the factors of 6 are 1, 2, 3, and 6.

b. Multiples: A multiple of an integer n is the product of n and another integer. For example, the first five multiples of 3 are 3, 6, 9, 12, and 15.

Understanding the properties of integers, divisibility rules, and the concepts of factors and multiples are essential for solving GMAT problems related to integers. In the following parts, we will discuss additional concepts and properties of integers, such as prime numbers, least common multiple (LCM), greatest common divisor (GCD), and integer operations.

## GMAT Integers – Number Properties (Part 2)

### Prime Numbers

Prime numbers are integers greater than 1 that have only two distinct positive factors: 1 and the number itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, and 17. The number 1 is not considered a prime number because it only has one distinct positive factor. The number 2 is unique, as it is the only even prime number.

### Composite Numbers

Composite numbers are integers greater than 1 that are not prime numbers. These numbers have more than two distinct positive factors. For example, the first few composite numbers are 4, 6, 8, 9, 10, and 12.

### Least Common Multiple (LCM)

The least common multiple of two or more integers is the smallest positive integer that is a multiple of each of the given integers. LCM can be found using the prime factorization method or by listing the multiples of each integer and finding the smallest common multiple.

Example: Find the LCM of 12 and 15.

Method 1: Prime Factorization 12 = 2^2 × 3^1 15 = 3^1 × 5^1

LCM(12, 15) = 2^2 × 3^1 × 5^1 = 60

Method 2: Listing Multiples Multiples of 12: 12, 24, 36, 48, 60, … Multiples of 15: 15, 30, 45, 60, …

LCM(12, 15) = 60

### Greatest Common Divisor (GCD)

The greatest common divisor (also known as the greatest common factor) of two or more integers is the largest positive integer that divides each of the given integers without leaving a remainder. GCD can be found using the prime factorization method or by listing the factors of each integer and finding the largest common factor.

Example: Find the GCD of 18 and 24.

Method 1: Prime Factorization 18 = 21 × 32 and 24 = 23 × 31

GCD(18, 24) = 21 × 31 = 6

Method 2: Listing Factors Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

GCD(18, 24) = 6

Understanding prime and composite numbers, along with the concepts of LCM and GCD, is crucial for solving advanced GMAT Inregers problems related to integers. In the next parts, we will discuss additional properties of integers, such as absolute value, consecutive integers, and remainders.

## GMAT Integers – Number Properties (Part 3)

### Absolute Value

The absolute value of an integer is the distance between the integer and zero on the number line, regardless of the integer’s sign. The absolute value is always non-negative. It is denoted by two vertical bars enclosing the integer, like this: |x|. For example, |5| = 5 and |-5| = 5.

### Consecutive Integers

Consecutive integers are integers that follow each other in order. For example, 3, 4, and 5 are consecutive integers. Consecutive even and odd integers can also be considered, such as 2, 4, 6 or 1, 3, 5. When solving GMAT problems, consecutive integers can be represented algebraically as x, x+1, x+2, and so on.

### Remainders

When dividing integers, sometimes there is a remainder. The remainder is the portion left over after division. For example, when dividing 7 by 3, the quotient is 2, and the remainder is 1 (since 7 = 3 × 2 + 1).

The remainder concept is often used in GMAT problems to test divisibility, modular arithmetic, and pattern recognition. Familiarity with the remainder theorem and the division algorithm is essential for solving these types of problems.

### Modular Arithmetic

Modular arithmetic is a system of arithmetic that considers the remainder when dividing integers. In modular arithmetic, two numbers are said to be congruent modulo n if they have the same remainder when divided by n. This is denoted as a ≡ b (mod n).

Example: 17 ≡ 5 (mod 6), since both 17 and 5 have a remainder of 5 when divided by 6.

Modular arithmetic is useful for solving GMAT problems that involve cycles, patterns, and remainders.

Understanding absolute values, consecutive integers, remainders, and modular arithmetic will strengthen your ability to tackle challenging GMAT problems related to integers. In the final part, we will discuss additional integer properties, such as integer operations and inequalities.

## GMAT Integers – Number Properties(Part 4)

### Integer Operations

When solving GMAT problems, it is essential to understand how integer operations affect the outcome:

Adding two even integers or two odd integers will always result in an even integer.

Adding one even integer and one odd integer will always result in an odd integer.

Subtracting two even integers or two odd integers will always result in an even integer.

Subtracting one even integer from an odd integer or vice versa will always result in an odd integer.

### b. Multiplying:

Multiplying two even integers, two odd integers, or one even integer and one odd integer will always result in an even integer.

Multiplying an even integer and an odd integer will always result in an even integer.

### Integer Inequalities

Inequalities involving integers can be solved by applying the same basic principles as for equalities:

a. Adding or subtracting the same integer to both sides of an inequality does not change the inequality’s direction. b. Multiplying or dividing both sides of an inequality by a positive integer does not change the inequality’s direction. c. Multiplying or dividing both sides of an inequality by a negative integer reverses the inequality’s direction.

Example: Solve the inequality -3x < 12.

Divide both sides by -3 and reverse the inequality’s direction: x > -4.

Integer Exponents

When working with integer exponents, it is essential to understand the following properties:

a. Any non-zero integer raised to the power of zero is 1. b. Any non-zero integer raised to the power of 1 is the integer itself. c. For any integers a and b, ab × ac = a(b+c) and (ab)c = a(b×c).

Understanding the properties of integer operations, inequalities, and exponents is crucial for solving advanced GMAT problems related to integers. By mastering these concepts, you will be well-prepared to tackle a wide range of integer problems on the GMAT.

## Integers Practice questions with an answer key.

Practice Question 1:

Which of the following integers is a multiple of both 4 and 6?

A) 12 B) 16 C) 24 D) 36 E) 48

Practice Question 2:

If x is an even integer, which of the following must also be even?

A) x + 1 B) x – 3 C) x × 2 D) x / 2 E) x + 3

Practice Question 3:

If |x| = 5, which of the following could be the value of x?

A) -5 B) 0 C) 2 D) 4 E) 7

Practice Question 4:

What is the greatest common divisor (GCD) of 18 and 30?

A) 2 B) 3 C) 6 D) 9 E) 18

Practice Question 5:

Which of the following integers is a prime number?

A) 4 B) 6 C) 8 D) 9 E) 11

Practice Question 6:

What is the least common multiple (LCM) of 10, 12, and 15?

A) 60 B) 120 C) 180 D) 240 E) 360

Practice Question 7:

Which of the following expressions is equal to |3x – 2| + |2x + 3|?

A) |5x + 1| B) |5x – 1| C) |x – 5| D) |x + 5| E) |x|

Practice Question 8:

How many integers between 10 and 100 are divisible by both 3 and 4?

A) 7 B) 8 C) 9 D) 10 E) 11

Practice Question 9:

If x and y are consecutive odd integers, which of the following must be divisible by 8?

A) x + y B) x – y C) x × y D) x2 + y2 E) x3 + y3

Practice Question 10:

If n is a positive integer and 2n > 100, what is the smallest possible value of n?

A) 6 B) 7 C) 8 D) 9 E) 10

Practice Question 11:

If a and b are positive integers such that ab = 36, what is the least possible sum of a and b?

A) 13 B) 14 C) 15 D) 16 E) 17

Practice Question 12:

If n is an integer, which of the following is NOT always an integer?

A) 2n + 3 B) n2 – 4 C) 5n/2 D) 3n – 5 E) n(n + 1)

Practice Question 13:

For how many positive integers n is the fraction (2n – 1)/n a prime number?

A) 1 B) 2 C) 3 D) 4 E) 5

Practice Question 14:

Let S be the sum of all distinct positive integers that are factors of 450. What is the sum of the digits of S?

A) 18 B) 21 C) 24 D) 27 E) 30

Practice Question 15:

If x is a positive integer and the greatest common divisor (GCD) of x and 54 is 9, what is the least possible value of x?

A) 9 B) 18 C) 27 D) 36 E) 45

Practice Question 16:

If a and b are positive integers and a3 × b = 72,000, what is the smallest possible value of a + b?

A) 25 B) 29 C) 32 D) 35 E) 37

Practice Question 17:

For how many integers n between 1 and 200 (inclusive) is 5n + 5 divisible by 25?

A) 49 B) 50 C) 51 D) 52 E) 53

Practice Question 18:

If the sum of the first n positive integers is 1,820, what is the value of n?

A) 60 B) 61 C) 62 D) 63 E) 64

Practice Question 19:

Let S be the product of all distinct prime numbers less than 20. What is the sum of the digits of S?

A) 9 B) 10 C) 12 D) 14 E) 16

Practice Question 20:

If x is an integer, which of the following expressions can be written as the difference of two squares?

A) x3 – 1 B) x3 + x2 C) x4 – 1 D) x4 + 2x2 + 1 E) x4 + x2 + 1

#### For each Data sufficiency Question, the Answer choice would be

###### E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Practice Question 1:

Is the integer x divisible by 3?

(1) x is divisible by 9.

(2) x is divisible by 6.

Practice Question 2:

What is the value of integer n?

(1) n is a prime number.

(2) n is an odd number greater than 10.

Practice Question 3:

Is the product of integers a and b even?

(1) a is an odd number.

(2) b is an even number.

Practice Question 4:

Is the integer x a multiple of 12?

(1) x is a multiple of 3.

(2) x is a multiple of 4.

Practice Question 5:

What is the value of x3 – y3?

(1) x – y = 3

(2) x2 + xy + y2 = 91

Practice Question 6:

Is the integer n even?

(1) n2 is even.

(2) n + 1 is odd.

Practice Question 7:

Is the integer n divisible by 15?

(1) n is divisible by 5.

(2) n is divisible by 9.

Practice Question 8:

If x and y are integers, is x2 + y2 odd?

(1) x + y is odd.

(2) x – y is even.

Practice Question 9:

Is x2 – y2 divisible by 4?

(1) x + y is divisible by 4.

(2) x – y is divisible by 4.

Practice Question 10:

Is the positive integer x a perfect square?

(1) The number of positive divisors of x is odd.

(2) The sum of the positive divisors of x is even.

Practice Question 11:

If a and b are integers, what is the value of a – b?

(1) The greatest common divisor of a and b is 12.

(2) The least common multiple of a and b is 84.