Problem 20 · 2016 AMC 8
Medium
Number Theory
divisibilityfactorization
The least common multiple of a and b is 12, and the least common multiple of b and c is 15. What is the least possible value of the least common multiple of a and c?
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Answer: A — 20.
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Hint 1 of 2
b is the hinge between the two conditions, so pin it down first. Since b is a factor of lcm(a,b) = 12 AND of lcm(b,c) = 15, it must divide BOTH 12 and 15 — so b divides gcd(12, 15) = 3.
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Hint 2 of 2
To make lcm(a, c) as small as possible, you want a and c to share as much as they can with b and carry no extra junk. Take b = 3, then pick the SMALLEST a and c that still satisfy each lcm condition.
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Approach: pin down the shared value b, then minimize a and c
- b divides both lcms, so b | gcd(12, 15) = 3 — meaning b is 1 or 3. To let b absorb shared factors, try b = 3.
- Smallest a with lcm(a, 3) = 12 is a = 4 (since 4 supplies the 2² and 3 supplies the 3). Smallest c with lcm(c, 3) = 15 is c = 5.
- lcm(4, 5) = 20 — and 4 and 5 share no factors, so this is as small as it gets: 20.
- Watch the trap: the lazy choice b = 1 forces a = 12, c = 15, giving lcm = 60 (answer C). Letting b carry the common 3 is what shrinks the answer to 20.
- Why this transfers: a value shared across two lcm/gcd conditions must divide the gcd of the two results — nail that shared value first, then minimize the leftovers.
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