Important Concepts: Factors and Multiples | Quantitative Reasoning for GMAT PDF Download

Introduction

Factors and multiples are closely connected ideas in number theory. A factor (or divisor) of a number is a whole number that divides that number exactly, leaving no remainder. A multiple of a number is the result obtained when that number is multiplied by an integer. If p × q = z, then z is a multiple of both p and q, and p and q are factors of z. For example, in 6 × 2 = 12, 6 and 2 are factors of 12, while 12 is a multiple of 6 and 2.

How to Find Factors and Multiples

Finding Factors of a Number

To determine the factors of a number, list the whole numbers that divide it exactly (without remainder). Every number has at least two factors: 1 and the number itself. The set of factors of a number is finite.

Steps to find factors (illustrated with 28)

• List pairs of whole numbers whose product equals the given number.
• For 28, consider pairs such as 1 × 28, 2 × 14 and 4 × 7.
• Each number in these pairs divides 28 exactly, so each is a factor.
• Collect all distinct numbers from these pairs to get the full list of factors.
• Thus the factors of 28 are 1, 2, 4, 7, 14 and 28.

Note: If a number is prime, its only factors are 1 and itself. If it is composite, it has more than two factors.

Finding Multiples of a Number

A multiple of a number is obtained by multiplying that number by whole numbers (0, 1, 2, 3, ...). The set of multiples of any non-zero integer is infinite. Multiples are useful for skip counting and for solving problems involving repeated addition or scheduling.

For example, the multiples of 28 obtained by multiplying 28 with whole numbers are shown in skip-count order below.

From the image and by calculation, some multiples of 28 are 28, 56, 84, 112, 140, 168, ...

Common Factors and Common Multiples

Common factors of two or more numbers are the factors that appear in each of their factor lists. Common multiples of two or more numbers are numbers that are multiples of each of them. Common factors are finite in number; common multiples are infinite.

Method to find common factors

• Write each given number.
• List all possible factors for each number.
• Identify the numbers that appear in every list-these are the common factors.
• Circle or note the common factors for clarity.

Example: Common factors of 30 and 42

Therefore, the common factors of 30 and 42 are 1, 2, 3 and 6.

Example: Common factors of 42, 70 and 84

Hence, the common factors of 42, 70 and 84 are 1, 2, 7 and 14.

Methods to find common multiples

• Listing multiples method: list multiples of each number and find the intersection.
• Venn diagram method: mark multiples of each number in separate regions and read the intersection for common multiples.
• Prime-factor method (useful for finding least common multiple): express each number as a product of prime powers and combine the highest powers.

There are infinitely many common multiples of any two non-zero integers. The smallest positive common multiple is called the least common multiple (LCM).

To illustrate the Venn diagram idea for multiples, consider multiples of 3 and 4. The multiples that occur in both sets are the common multiples. The first three positive common multiples of 3 and 4 are 12, 24 and 36.

Listing multiples example: 2 and 4

• Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...
• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
• Common multiples of 2 and 4: 4, 8, 12, 16, 20, 24, ...

If one number is a factor of another (for example 2 is a factor of 4), then every multiple of the larger number is also a common multiple.

Properties of Factors and Multiples

Key properties to remember:

• Every whole number has 1 as a factor.
• Every number is a multiple of itself.
• Zero multiplied by any number gives 0, so 0 is a multiple of every number.
• Each number has a finite set of factors but infinitely many multiples.
• A number with exactly two distinct positive factors (1 and itself) is a prime number; otherwise it is composite.
• If a divides b (that is, a is a factor of b), then every multiple of b is also a multiple of a.
• The greatest common divisor (GCD) or greatest common factor (GCF) is the largest number that divides each of the given numbers.
• The least common multiple (LCM) is the smallest positive integer that is a multiple of each of the given numbers.
• For two positive integers a and b, the relation LCM(a, b) × GCD(a, b) = a × b holds.

Important Notes

• Multiples of any non-zero integer are infinite, so any pair of non-zero integers have infinitely many common multiples.
• The LCM is the least (smallest positive) common multiple. For example, LCM(3, 4) = 12.
• The GCD is useful for simplifying fractions; the LCM is useful for finding a common denominator when adding or subtracting fractions and for solving synchronisation/scheduling problems.
• Common computational methods include: listing, prime factorisation, the Euclidean algorithm for GCD, and using prime factors to compute LCM.

Solved Examples

Example 1: Find the common factors and multiples of 4, 8, and 12.
Ans:

To find the common factors and multiples of 4, 8, and 12 we will use the listing method.

Factors of 4:
1, 2 and 4.
Factors of 8:
1, 2, 4 and 8.
Factors of 12:
1, 2, 3, 4, 6 and 12.
Common factors are the numbers that appear in all three factor lists.
1, 2 and 4 are common factors of 4, 8 and 12.

Multiples of 4:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, ...
Multiples of 8:
8, 16, 24, 32, 40, 48, 56, 64, 72, 80, ...
Multiples of 12:
12, 24, 36, 48, 60, 72, ...
Common multiples are the numbers that occur in all three lists.
Thus the common multiples of 4, 8 and 12 are 24, 48, 72, ...

Example 2: What are the factors and multiples of 60?
Ans:

Factors of 60 are all whole numbers that divide 60 exactly.
List factors by considering factor pairs of 60:
1 × 60
2 × 30
3 × 20
4 × 15
5 × 12
6 × 10
Collect the distinct numbers from these pairs.
Therefore, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

To list multiples of 60, multiply 60 by consecutive whole numbers starting from 1.
60 × 1 = 60
60 × 2 = 120
60 × 3 = 180
60 × 4 = 240
60 × 5 = 300
Thus multiples of 60 include 60, 120, 180, 240, 300, ...

Additional Methods and Applications

Prime factorisation method (for factors, GCD and LCM): express each number as a product of prime powers and use those to compute GCD and LCM.

• To find the GCD, take the product of common prime factors with the smallest exponents appearing in each factorisation.
• To find the LCM, take the product of all prime factors with the largest exponent appearing in any factorisation.

Euclidean algorithm (efficient method for GCD): repeatedly divide and replace until the remainder is zero. The last non-zero remainder is the GCD.

Applications: simplifying fractions, finding common denominators, solving problems involving schedules and cycles (when events repeat after fixed intervals), ratio problems, and number-theoretic questions such as divisibility and modular reasoning.

Summary

Factors are divisors of a number and are finite in number; multiples arise from multiplying by whole numbers and are infinite. Use listing, prime factorisation, Venn diagrams, or the Euclidean algorithm to find common factors and multiples. Remember the useful relation LCM(a, b) × GCD(a, b) = a × b and apply these concepts when simplifying fractions, finding common denominators or solving periodic/scheduling problems.