By Claire Gillespie, Updated Aug 30, 2022

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Understanding the factors of a number is essential for mastering basic arithmetic, algebra, and calculus. A factor is any integer that divides a number exactly, including 1 and the number itself.

TL;DR

To find all factors quickly, keep dividing the number by its smallest prime divisor until you reach 1. Record each divisor; the collection of these gives the full factor list.

Prime Numbers

A prime number is divisible only by 1 and itself. Common examples are 2, 3, 5, 7, 11, and 13. The integer 1 is not prime, because it divides every number.

Divisibility Rules

Several simple rules help identify factors:

• Even numbers are divisible by 2.
• If the sum of a number's digits is a multiple of 3, the number is divisible by 3.
• Numbers ending in 0 or 5 are divisible by 5.
• Being divisible by 2 twice indicates divisibility by 4.
• Divisibility by both 2 and 3 implies divisibility by 6.
• Divisibility by 3 twice (or a digit sum divisible by 9) means divisibility by 9.

Finding Factors Quickly

Start with the number you wish to factor, e.g., 24. Write down pairs that multiply to 24:

• 1 × 24
• 2 × 12
• 3 × 8
• 4 × 6

Thus, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Negative numbers follow the same logic, but the product of the factors must be negative. For -30, the factor pairs are -1 × 30, 1 × -30, -2 × 15, 2 × -15, -3 × 10, 3 × -10, -5 × 6, and 5 × -6.

For larger numbers, a systematic table helps. Take 3,784 as an example:

• 2 × 1,892 (write 2 in the left column, 1,892 on the right)
• 2 × 946 (add 2 and 946 to the table)
• 2 × 473 (now 473 is odd; continue with the next smallest prime)
• 11 × 43 (473 = 11 × 43)

Continue this process until the right-hand number equals 1. The complete list of prime factors is 2, 2, 2, 11, and 43, and the full set of composite factors can be built from these primes.