Introduction and Basic Definitions

I. Introduction

In this four-part series on Factors and Multiples – Number Properties, we will explore the concepts of factors and multiples, their properties, and how these ideas are used in solving GMAT quantitative problems. A strong understanding of factors and multiples is essential for acing the GMAT and performing well in the Quantitative section.

Part 1 will cover the basic definitions and examples of factors and multiples. In subsequent parts, we will delve deeper into the properties of these concepts, learn techniques to identify common factors and multiples, and explore problem-solving strategies related to these topics.

II. Factors

A factor is a number that divides another number completely, leaving no remainder. In other words, if a number ‘a’ can be divided by another number ‘b’ without any remainder, then ‘b’ is a factor of ‘a’. For example, the factors of 6 are 1, 2, 3, and 6, since these numbers divide 6 completely without leaving any remainder.

Types of Factors

a. Prime Factors: These are the factors that are prime numbers themselves, meaning they can only be divided by 1 and themselves. For example, the prime factors of 12 are 2 and 3, as 2² ×  3 = 12.

b. Composite Factors: These are not prime factors, meaning they can be divided by numbers other than one and themselves. For example, the composite factors of 12 are 4 and 6, as 4 = 2 ×  2 and 6 = 2  3.

III. Multiples

A multiple is a number that can be obtained by multiplying a given number by an integer. In other words, if ‘a’ is a multiple of ‘b’, then ‘a’ can be written as a product of ‘b’ and an integer. For example, the first five multiples of 6 are 6, 12, 18, 24, and 30, as these numbers can be obtained by multiplying 6 by integers 1, 2, 3, 4, and 5, respectively.

Types of Multiples

a. Common Multiples: These are the multiples that two or more numbers share. For example, the common multiples of 3 and 4 are 12, 24, 36, and so on, as these numbers can be obtained by multiplying both 3 and 4 by some integer.

b. Least Common Multiple (LCM): The smallest multiple that two or more numbers share is called the Least Common Multiple. For example, the LCM of 3 and 4 is 12, as it is the smallest number that can be obtained by multiplying both 3 and 4 by some integer.

In the next part, we will discuss the properties of factors and multiples and learn how to find the prime factorization of a number, which is crucial for solving many GMAT problems.

Properties of Factors and Multiples and Prime Factorization

I. Properties of Factors and Multiples

Every number has at least two factors: itself and one. For example, the factors of 7 are 1 and 7.

Every number has at least two multiples: itself and zero. For example, the multiples of 7 are 0, 7, 14, and so on.

The number 1 is a factor of every number, but the only multiple of 1 is itself.

The smallest multiple of any number is zero.

If a number is a multiple of another number, the factors of the smaller number are also factors of the larger number. For example, if 6 is a multiple of 3 (6 = 3 ×  2), then the factors of 3 (1 and 3) are also factors of 6.

If a number is a factor of another number, the multiples of the larger number are also multiples of the smaller number. For example, if 3 is a factor of 6 (6 = 3 ×  2), then the multiples of 6 (6, 12, 18, etc.) are also multiples of 3.

II. Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors. This is an essential technique for solving many GMAT problems related to factors and multiples.

Steps for Prime Factorization

a. Start with the smallest prime number, 2, and determine if it divides the given number. If it does, divide the number by 2 and record the quotient.

b. Repeat this process with the quotient and the smallest prime number that divides it until the quotient is a prime number.

c. The prime factorization is the list of prime numbers that you used to divide the original number, along with the final prime quotient.

Example: Prime Factorization of 84

a. 84 is divisible by 2, the smallest prime number. 84 ÷ 2 = 42. Record the prime factor 2 and the quotient 42.

b. 42 is also divisible by 2. 42 ÷ 2 = 21. Record another prime factor 2 and the quotient 21.

c. 21 is not divisible by 2 but is divisible by the next smallest prime number, 3. 21 ÷ 3 = 7. Record the prime factor 3 and the quotient 7.

d. 7 is a prime number, so the process stops. The prime factorization of 84 is 2² ×  3 ×  7.

In Part 3, we will delve deeper into the concepts of Greatest Common Divisor (GCD) and Least Common Multiple (LCM) and their applications in solving GMAT problems.

Greatest Common Divisor (GCD) and Least Common Multiple (LCM)

I. Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF), is the largest number that divides two or more numbers without leaving a remainder. It is a useful concept for solving many GMAT problems.

Finding the GCD using Prime Factorization

a. Find the prime factorization of each number. b. Identify the common prime factors and their lowest powers. c. Multiply the common prime factors with their lowest powers.

Example: Finding the GCD of 36 and 48

a. Prime factorization of 36: 2² ×  3² Prime factorization of 48: 2⁴ ×  3

b. Common prime factors: 2 and 3 Lowest powers: 2² and 3¹

c. GCD(36, 48) = 2² ×  3¹ = 12

II. Least Common Multiple (LCM)

The Least Common Multiple (LCM) is the smallest positive number that is a multiple of two or more numbers. It is another essential concept for solving GMAT problems.

Finding the LCM using Prime Factorization

a. Find the prime factorization of each number. b. Identify all prime factors from both numbers and their highest powers. c. Multiply all prime factors with their highest powers.

Example: Finding the LCM of 36 and 48

a. Prime factorization of 36: 2² ×  3² Prime factorization of 48: 2⁴ ×  3

b. All prime factors: 2 and 3 Highest powers: 2⁴ and 3²

c. LCM(36, 48) = 2⁴ ×  3² = 144

III. Relationship between GCD and LCM

For any two positive integers a and b:

GCD(a, b) ×  LCM(a, b) = a ×  b

This relationship is useful when solving problems that involve both the GCD and LCM.

Problem-Solving Strategies and Techniques

I. Identifying Prime Numbers

A prime number is a number greater than 1 that has exactly two factors: 1 and itself. To check if a number is prime, follow these steps:

Find the square root of the number.

Check for divisibility by prime numbers less than or equal to the square root.

If the number is not divisible by any of these prime numbers, it is prime.

Example: Is 29 a prime number?

√29 ≈ 5.4

Check divisibility by 2, 3, and 5 (prime numbers less than or equal to 5.4).

29 is not divisible by 2, 3, or 5, so it is a prime number.

II. Finding the Number of Factors

To find the number of factors for a given number, follow these steps:

Find the prime factorization of the number.

Add 1 to each exponent in the prime factorization.

Multiply the resulting numbers together.

Example: Find the number of factors of 60.

Prime factorization of 60: 2² ×  3¹ ×  5¹

Add 1 to each exponent: (2 + 1) ×  (1 + 1) ×  (1 + 1)

Multiply the resulting numbers: 3 ×  2 ×  2 = 12 factors

III. Determining the Sum of Factors

To find the sum of all factors of a given number, follow these steps:

Find the prime factorization of the number.

For each prime factor, raise it to one power greater than its exponent and subtract one. Then, divide the result by the prime factor minus one.

Multiply the resulting numbers together.

Example: Find the sum of factors of 60.

Prime factorization of 60: 2² ×  3¹ ×  5¹

Apply the formula to each prime factor:

For 2: (2⁽²⁺¹⁾ – 1) / (2 – 1) = (2³ – 1) / 1 = 7

For 3: (3⁽¹⁺¹⁾ – 1) / (3 – 1) = (3² – 1) / 2 = 4

For 5: (5⁽¹⁺¹⁾ – 1) / (5 – 1) = (5² – 1) / 4 = 6

Multiply the resulting numbers: 7 ×  4 ×  6 = 168

In conclusion, factors and multiples are fundamental concepts in number properties, and a thorough understanding of these topics is crucial for acing the GMAT Quantitative section. By mastering the techniques presented in this four-part series, you will be well-equipped to tackle GMAT problems related to factors and multiples.