Euclid Algorithm for HCF

Quick answer: Euclid's algorithm repeatedly replaces two positive integers by the smaller number and the remainder. The last nonzero remainder is their HCF, because common divisors are preserved at each step.

Worked example

Find the HCF of 252 and 105.

1. 252 = 2×105 + 42.
2. 105 = 2×42 + 21.
3. 42 = 2×21 + 0.
4. The last nonzero remainder is 21, so HCF = 21.

Check the result

252/21 = 12 and 105/21 = 5; the quotients 12 and 5 are coprime.

Common mistake to avoid

The HCF is not the final zero remainder. Keep dividing until the remainder is zero, then take the previous divisor.

Try it yourself

1. Find HCF(48,18). 2. Find HCF(35,64).

Answers and reasoning

1. 48 = 2×18 + 12; 18 = 12 + 6; 12 = 2×6, so HCF 6. 2. Remainders are 29, 6, 5, 1, 0, so HCF 1.

Need help with your own attempt? Ask a study question. AI answers require an account and paid credits. Check important steps with your textbook or teacher.