What Are the Factors of 84?
Eighty-four months is exactly 7 years, and 84 also splits into 12 groups of 7 or 4 groups of 21. Every one of those tidy splits is a factor: a whole number that divides another number exactly, with no remainder.
The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. That is twelve factors.
Factor Pairs of 84
Look for pairs that multiply to 84, working from the outside in.
- Start with 1 × 84.
- Try 2. Since 84 ÷ 2 = 42, write 2 × 42.
- Try 3. Since 84 ÷ 3 = 28, write 3 × 28.
- Try 4. Since 84 ÷ 4 = 21, write 4 × 21.
- Try 6. Since 84 ÷ 6 = 14, write 6 × 14.
- Try 7. Since 84 ÷ 7 = 12, write 7 × 12.
- Keep trying whole numbers until you reach 12, which is already in your list. Every factor is now found, so you are done.
Prime Factorization of 84
Prime factors show what 84 is built from. Split it and keep going until every branch is a prime.
So 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7. Its prime factors are 2, 3 and 7. Add 1 to each exponent and multiply: (2 + 1) × (1 + 1) × (1 + 1) = 12, which matches the list.
Divisibility check for 84
A number divides by 84 when it divides by 4, 3 and 7. Take 252: its last two digits give 52 ÷ 4 = 13, its digits add up to 9, and 252 ÷ 7 = 36, so 252 is a multiple of 84.
Is 84 a Prime or Composite Number?
Composite. It has twelve factors. The sum of all factors is 1 + 2 + 3 + 4 + 6 + 7 + 12 + 14 + 21 + 28 + 42 + 84 = 224. The factors smaller than 84 add up to 140, which is more than 84, so 84 is called an abundant number.
Real-Life Uses of the Factors of 84
- Years. Eighty-four months is exactly 7 years.
- Dozens. Eighty-four items make 7 dozen.
- Rows. Eighty-four chairs fit in 7 rows of 12 or 6 rows of 14.
Common Factors of 84 With Other Numbers
The factors of 126 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63 and 126, so 84 and 126 share 1, 2, 3, 6, 7, 14, 21 and 42. The GCF is 42, which simplifies 84/126 to 2/3.
Quick Practice
Which of these are factors of 84: 5, 6 or 9? Only 6. Now try 90 and check it with the calculator.