Find the HCF directly: take each prime the numbers share, the fewest times it appears. Works for three numbers too.
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The HCF uses only shared primes, each the fewest times it appears.
We want the largest common subpart. Build it prime by prime:
HCF of 30 and 72, and of 225 and 750, without listing factors.
Problem. Find the HCF of 30 and 72, then of 225 and 750. (Textbook pp. 53–54, Examples 4 and 5)
Common primes only, each the fewest times.
Fill in the prime counts to find the HCF.
Figure it Out, Textbook p. 54, Q1(a).
24 = 2 × 2 × 2 × 3 and 180 = 2 × 2 × 3 × 3 × 5.
Common primes: 2 and .
Fewest 2s: . Fewest 3s: .
HCF = 2 × 2 × 3 = .
HCF of the book's number sets using the minimum rule.
HCF of 42, 75 and 24?
Why a composite factorisation can hide common factors.
Figure it Out, Textbook p. 54, Q2. 72 and 144 are factorised into composite numbers as and . No factor appears in both. Seeing this, can we say the two numbers have no common factor other than 1? Why not?
Check each.
Use the division method.
Use the minimum rule.
Explain why the composite form misled us.