What Are the Factors of 105?
A year group of 105 pupils could be split into 3 groups of 35, 5 groups of 21 or 7 groups of 15. Every one of those tidy splits is a factor: a whole number that divides another number exactly, with no remainder.
The factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105. That is eight factors.
Factor Pairs of 105
Look for pairs that multiply to 105, working from the outside in.
- Start with 1 × 105.
- Try 3. Since 105 ÷ 3 = 35, write 3 × 35.
- Try 5. Since 105 ÷ 5 = 21, write 5 × 21.
- Try 7. Since 105 ÷ 7 = 15, write 7 × 15.
- Keep trying whole numbers until you reach 15, which is already in your list. Every factor is now found, so you are done.
Prime Factorization of 105
Prime factors show what 105 is built from. Split it and keep going until every branch is a prime.
So 105 = 3 × 5 × 7. Its prime factors are 3, 5 and 7. Each prime appears once, so the count is (1 + 1) × (1 + 1) × (1 + 1) = 8, which matches the list.
Divisibility check for 105
A number divides by 105 when it divides by 3, 5 and 7. Take 315: it ends in 5, its digits add up to 9, and 315 ÷ 7 = 45, so 315 is a multiple of 105.
Is 105 a Prime or Composite Number?
Composite. It has eight factors. The sum of all factors is 1 + 3 + 5 + 7 + 15 + 21 + 35 + 105 = 192. The factors smaller than 105 add up to 87, which is less than 105, so 105 is called a deficient number.
Real-Life Uses of the Factors of 105
- Pupils. A year group of 105 splits into 3 groups of 35 or 5 groups of 21.
- Weeks. 105 days is exactly 15 weeks.
- Rows. 105 seats fit in 7 rows of 15.
Common Factors of 105 With Other Numbers
The factors of 135 are 1, 3, 5, 9, 15, 27, 45 and 135, so 105 and 135 share 1, 3, 5 and 15. The GCF is 15, which simplifies 105/135 to 7/9.
Quick Practice
Which of these are factors of 105: 6, 5 or 9? Only 5. Now try 112 and check it with the calculator.