2021 AMC 12A Fall Problem 8

Below is the professionally curated solution for Problem 8 of the 2021 AMC 12A Fall, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Fall solutions, or check the answer key.

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Concepts:least common multiple prime factorization

Difficulty rating: 1440

Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M, 32, 33, 34, 35, 36, 37, 38, 39,$ and $40.$ What is the value of $\dfrac{N}{M}?$

$1$

$2$

$37$

$74$

$2886$

Solution:

$M = \operatorname{lcm}(10, \ldots, 30)$ contains $2^4$ (from $16$ ), $3^3$ (from $27$ ), $5^2$ (from $25$ ), $7,$ and every prime up to $29.$

Among $32, \ldots, 40,$ the only new contributions are $32 = 2^5,$ which raises the power of $2$ from $2^4$ to $2^5,$ and the new prime $37.$ Everything else factors into primes and powers already in $M.$

Therefore $\dfrac{N}{M} = 2 \cdot 37 = 74.$

Thus, the correct answer is D.

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