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2013 AMC 8

Problem 10

Problem 10 · 2013 AMC 8 Medium
Number Theory lcm-gcdprime-factorization

What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?

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Answer: C — 330.
Show hints
Hint 1 of 2
Don't compute the huge LCM and tiny GCD separately and then divide. Once both numbers are in prime-factored form, the ratio reads off prime by prime: for each prime, LCM takes the bigger exponent and GCD the smaller, so the ratio just keeps the difference.
Still stuck? Show hint 2 →
Hint 2 of 2
Prime factorization is the master key for LCM and GCD: GCD = product of min exponents, LCM = product of max exponents. Compare the two prime towers side by side.
Show solution
Approach: prime factor each number, then compare exponents
  1. Factor: 180 = 22 · 32 · 5 and 594 = 2 · 33 · 11.
  2. GCD takes the smaller power of each shared prime: 21 · 32 = 18.
  3. LCM takes the larger power of every prime that appears: 22 · 33 · 5 · 11 = 4 · 27 · 55 = 5940.
  4. Ratio: 5940 ÷ 18 = 330.
Another way — ratio straight from exponent gaps (no big numbers):
  1. LCM/GCD keeps, for each prime, the gap between the two exponents: 2 has exponents 2 and 1 (gap 1), 3 has 2 and 3 (gap 1), and 5, 11 appear in only one number (full power).
  2. So the ratio = 21 · 31 · 5 · 11 = 2 · 3 · 5 · 11 = 330 — no 5940 or 106,920 ever needed.
Another way — product identity LCM · GCD = a · b:
  1. Use LCM(a,b) · GCD(a,b) = a · b, so LCM/GCD = ab/GCD2.
  2. 180 · 594 = 106,920 and GCD = 18, giving 106,920 ÷ 324 = 330.
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