What Are the Factors of 72?
Seventy-two hours is exactly 3 days, and 72 also splits into 6 groups of 12 or 8 groups of 9. Every one of those tidy splits is a factor: a whole number that divides another number exactly, with no remainder.
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72. That is twelve factors.
Factor Pairs of 72
Look for pairs that multiply to 72, working from the outside in.
- Start with 1 × 72.
- Try 2. Since 72 ÷ 2 = 36, write 2 × 36.
- Try 3. Since 72 ÷ 3 = 24, write 3 × 24.
- Try 4. Since 72 ÷ 4 = 18, write 4 × 18.
- Try 6. Since 72 ÷ 6 = 12, write 6 × 12.
- Try 8. Since 72 ÷ 8 = 9, write 8 × 9.
- Keep trying whole numbers until you reach 9, which is already in your list. Every factor is now found, so you are done.
Prime Factorization of 72
Prime factors show what 72 is built from. Split it and keep going until every branch is a prime.
So 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3². Its prime factors are 2 and 3. Add 1 to each exponent and multiply: (3 + 1) × (2 + 1) = 12, which matches the list.
Divisibility check for 72
A number divides by 72 when it divides by 8 and 9. Take 216: its last three digits give 216 ÷ 8 = 27, and its digits add up to 9, so 216 is a multiple of 72.
Is 72 a Prime or Composite Number?
Composite. It has twelve factors. The sum of all factors is 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 + 72 = 195. The factors smaller than 72 add up to 123, which is more than 72, so 72 is called an abundant number.
Real-Life Uses of the Factors of 72
- Hours. Seventy-two hours is exactly 3 days.
- Eggs. Seventy-two eggs make 6 dozen.
- Rows. Seventy-two tiles fit in 8 rows of 9 or 6 rows of 12.
Common Factors of 72 With Other Numbers
The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 90, so 72 and 90 share 1, 2, 3, 6, 9 and 18. The GCF is 18, which simplifies 72/90 to 4/5.
Quick Practice
Which of these are factors of 72: 5, 6 or 10? Only 6. Now try 84 and check it with the calculator.