Problem 17 · 2011 AMC 8
Medium
Number Theory
prime-factorization
Let w, x, y, and z be whole numbers. If 2w · 3x · 5y · 7z = 588, then what does 2w + 3x + 5y + 7z equal?
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Answer: A — 21.
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Hint 1 of 2
The bases 2, 3, 5, 7 are all prime, so a number's prime factorization tells you each exponent directly — just factor 588.
Still stuck? Show hint 2 →
Hint 2 of 2
Pull out the small primes one at a time (is it even? divisible by 3? by 7?) to find how many of each it contains.
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Approach: match prime powers — a prime factorization is unique, so the exponents are forced
- Factor 588 by peeling off small primes: 588 = 2 · 294 = 2 · 2 · 147 = 4 · 3 · 49 = 22 · 31 · 72.
- Match exponents on each prime: w = 2, x = 1, z = 2. There's no factor of 5, so y = 0 (and 50 = 1 contributes nothing — the easy slip is to forget this term).
- 2w + 3x + 5y + 7z = 4 + 3 + 0 + 14 = 21.
- Why this works: every whole number has exactly one prime factorization, so once both sides are written as products of primes, the exponents must match up one-for-one.
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