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Numbers are the language of mathematics and like any language, they have fundamental components. One of the most basic and crucial concepts in understanding numbers is that of factors. Think of factors as the "ingredients" or "building blocks" that, when multiplied together, form a larger number.

The core definition of a factor, as highlighted by our keyword, is: "If a number (or quantity/expression) is equal to the product of two or more other numbers (quantities/expressions), then each of these latter numbers is called a factor of the former number." In simpler terms, if you can multiply two whole numbers together to get a third number, then those two original numbers are factors of the third number.

What Exactly is a Factor?

When we say a number (let's call it 'A') is formed by multiplying two or more other numbers (let's call them 'B', 'C', etc.), such that A = B × C (or A = B × C × D), then 'B', 'C', and 'D' are all considered factors of 'A'.

For example, consider the factors of 12.

We know that 3 × 4 = 12. So, 3 and 4 are factors of 12.

We also know that 2 × 6 = 12. So, 2 and 6 are factors of 12. And 1 × 12 = 12. So, 1 and 12 are factors of 12.

Notice that each of these factors (1, 2, 3, 4, 6, 12) divides 12 evenly, without leaving any remainder.

N.B. 1 (one) is a Universal Factor and 0 (Zero) cannot be a factor of any number (division by zero is undefined). Every non-zero number is a factor of zero (e.g., 5 x 0 = 0).

Types of Factors

1. Prime Factors

Definition: A prime factor is a factor of a number that is also a prime number. A prime number is a whole number greater than 1 that has only two factors: 1 and itself (e.g., 2, 3, 5, 7, 11...).

Prime Factorization: Breaking down a number into its prime factors is called prime factorization. This is like finding the most basic "molecular" components of a number.

For 12, the prime factors are 2, 2, and 3 (because 2 × 2 × 3 = 12).

The factors of 12 are {1, 2, 3, 4, 6, 12}.

The prime numbers within this set are {2, 3}.

2. Common Factors

Definition: Common factors are the factors that two or more numbers share.

Example: Common Factors of 12 and 18.

Factors of 12: {1, 2, 3, 4, 6, 12}

Factors of 18: {1, 2, 3, 6, 9, 18}

The common factors of 12 and 18 are: {1, 2, 3, 6}.

3. Greatest Common Factor (GCF) or Highest Common Factor (HCF)

Definition: The GCF (or HCF) is the largest among the common factors of two or more numbers.

Example (continuing from above):

The common factors of 12 and 18 are {1, 2, 3, 6}.

The greatest among these is 6. So, the GCF of 12 and 18 is 6.

Importance: GCF is widely used for simplifying fractions and solving problems involving grouping or distribution.

Why factors are useful?

Table Of Factors 1-100