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What are two strategies we can use to find the gcf?
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GPT-5 mini
AI
Two common strategies are:
1. Prime factorization
- Factor each number into primes, find the primes they share, and multiply the lowest powers of those shared primes.
- Example: 48 = 2^4·3, 18 = 2·3^2 → common primes 2^1·3^1 = 6 → GCF = 6.
2. Euclidean algorithm (division method)
- Repeatedly divide: replace (a, b) by (b, a mod b) until the remainder is 0. The last nonzero remainder is the GCF.
- Example: gcd(48,18): 48 ÷ 18 = 2 r 12 → 18 ÷ 12 = 1 r 6 → 12 ÷ 6 = 2 r 0 → GCF = 6.
(For very small numbers you can also list all factors and pick the largest common one.)
1. Prime factorization
- Factor each number into primes, find the primes they share, and multiply the lowest powers of those shared primes.
- Example: 48 = 2^4·3, 18 = 2·3^2 → common primes 2^1·3^1 = 6 → GCF = 6.
2. Euclidean algorithm (division method)
- Repeatedly divide: replace (a, b) by (b, a mod b) until the remainder is 0. The last nonzero remainder is the GCF.
- Example: gcd(48,18): 48 ÷ 18 = 2 r 12 → 18 ÷ 12 = 1 r 6 → 12 ÷ 6 = 2 r 0 → GCF = 6.
(For very small numbers you can also list all factors and pick the largest common one.)
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