Stage 7 · Factors & multiples
Highest common factor
Short division: multiply the divisors.
How to do it
Method
Short division: multiply the divisors
24, 36 → 12, 18 → 6, 9 → 2, 3: 2 × 2 × 3 = 12
Another way
Short division, or listing
Common factors of 24 and 36: 1, 2, 3, 4, 6, 12 → 12
In full
Use short division: divide both numbers by a factor they share (try the primes 2, 3, 5, 7 first), and keep going until the two quotients share no factor but 1; the divisors multiplied are the H.C.F. 24 and 36: ÷ 2 → 12 and 18, ÷ 2 → 6 and 9, ÷ 3 → 2 and 3; 2 and 3 share nothing but 1, and 2 × 2 = 4, 4 × 3 = 12, so the H.C.F. is 12. Another way is listing: write out every factor of both numbers and pick the largest they share — the same answer.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
H.C.F. of 25 and 45 = ?
Show the answer and working
Answer: H.C.F. of 25 and 45 = 5
H.C.F. of 25 and 45: by short division, 25 and 45 ÷ 5 → 5 and 9; 5 and 9 share no factor but 1, so the H.C.F. is the divisor, 5
Example 2
H.C.F. of 14 and 12 = ?
Show the answer and working
Answer: H.C.F. of 14 and 12 = 2
H.C.F. of 14 and 12: by short division, 14 and 12 ÷ 2 → 7 and 6; 7 and 6 share no factor but 1, so the H.C.F. is the divisor, 2
Example 3
H.C.F. of 18 and 9 = ?
Show the answer and working
Answer: H.C.F. of 18 and 9 = 9
H.C.F. of 18 and 9: by short division, 18 and 9 ÷ 3 → 6 and 3; 6 and 3 ÷ 3 → 2 and 1; 2 and 1 share no factor but 1. Multiply the divisors: 3 × 3 = 9, so the H.C.F. is 9
Example 4
H.C.F. of 32 and 72 = ?
Show the answer and working
Answer: H.C.F. of 32 and 72 = 8
H.C.F. of 32 and 72: by short division, 32 and 72 ÷ 2 → 16 and 36; 16 and 36 ÷ 2 → 8 and 18; 8 and 18 ÷ 2 → 4 and 9; 4 and 9 share no factor but 1. Multiply the divisors: 2 × 2 = 4, 4 × 2 = 8, so the H.C.F. is 8
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