How can you quickly find the factors of a number?

Factors are the numbers that divide evenly into another number — with no remainder. A factor is always less than or equal to the original number.

To find all factors quickly:

  1. Start with 1 and the original number — these are always factors.
  2. Test each number starting from 2:
    • Ask: Does this number divide into the original without a remainder?
    • If yes, add both the divisor and the quotient to your list.
  3. Stop when you pass the square root of the number.

If you end with only 1 and the number itself, it’s prime.
If you find more than two factors, it’s composite.

Let’s work through this for the number 72.

Find Factors Of 72

Step-by-step:

  • Start with: $[1, 72]$
  • Try 2: $72 \div 2 = 36$ → ✅ → add 2 and 36
  • Try 3: $72 \div 3 = 24$ → ✅ → add 3 and 24
  • Try 4: $72 \div 4 = 18$ → ✅ → add 4 and 18
  • Try 5: $72 \div 5 = 14.4$ → ❌
  • Try 6: $72 \div 6 = 12$ → ✅ → add 6 and 12
  • Try 7: $72 \div 7 = 10.29\ldots$ → ❌
  • Try 8: $72 \div 8 = 9$ → ✅ → add 8 and 9

You’ve reached $\sqrt{72} \approx 8.48$, so stop here.

Final factor list:

$$ [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72] $$

  • Since 72 has more than two factors, it’s a composite number.
  • The highest factor (excluding 72) is 36.

Summary

The factors of 72 are:

$$ [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72] $$

  • A number with only 1 and itself as factors is a prime number.
  • A number with more than two factors is a composite number.

Since 72 has multiple divisors, it is clearly composite.