What is a Factor? Definition, Examples and Facts

What are Factors?

Factors of a number are integers that can be multiplied together to yield that specific number. Every whole number has at least two factors: 1 and the number itself. For instance, the factors of 6 are 1, 2, 3, and 6, as 1 × 6 = 6 and 2 × 3 = 6.

Characteristics of Factors

  1. Prime Factors:
    • Prime factors are factors that are prime numbers. The prime factorization of a number involves expressing it as the product of its prime factors.
  2. Factor Pairs:
    • Factors often come in pairs. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12, forming pairs such as (1, 12), (2, 6), and (3, 4).

How to Find Factors of a Number?

Finding Factors of a Number: A Step-by-Step Guide

Step 1: Start with 1 and the Number Itself

  • Every whole number has at least two factors: 1 and the number itself.

Step 2: Check for Small Factors

  • Begin with the smallest possible factor, 2.
  • Test if the number is divisible by 2. If yes, 2 is a factor.
  • Repeat this process with 3, 4, 5, and so on, until the quotient becomes greater than the divisor.

Step 3: Factor Pairs

  • As you identify factors, note them down.
  • Look for factor pairs. For example, if 3 is a factor, then \(\frac{n}{3}\) is also a factor.

Step 4: Stop at the Square Root

  • You only need to check factors up to the square root of the number.
  • If a number has a factor larger than its square root, it must also have a corresponding factor smaller than the square root.

Step 5: List All Factors

  • Compile a list of all the factors you have found.

Example: Finding Factors of 24

  1. Start with 1 and 24.
  2. Check for factors: 2, 3, 4, 6, and 8.
  3. Factor pairs: (1, 24), (2, 12), (3, 8), (4, 6).
  4. Stop at the square root of 24, which is approximately 4.9.
  5. List all factors: 1, 2, 3, 4, 6, 8, 12, and 24.

Tips:

  • Use divisibility rules for common divisors (e.g., divisibility by 2, 3, 5).
  • Be systematic in testing potential factors.
  • Factorization can be helpful in identifying factors quickly.

Examples of Factors

  1. Factors of 8:
    • The factors of 8 are 1, 2, 4, and 8.
    • Factor pairs: (1, 8) and (2, 4).
  2. Factors of 15:
    • The factors of 15 are 1, 3, 5, and 15.
    • Factor pairs: (1, 15) and (3, 5).