2019 AMC 12B Problem 5

Below is the professionally curated solution for Problem 5 of the 2019 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 12B solutions, or check the answer key.

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Concepts:least common multiple divisibility

Difficulty rating: 1200

Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either $12$ pieces of red candy, $14$ pieces of green candy, $15$ pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs $20$ cents. What is the smallest possible value of $n?$

$18$

$21$

$24$

$25$

$28$

Solution:

Let $M$ be Casper's money in cents. Since he can exactly buy $12,$ $14,$ or $15$ whole-cent pieces, $M$ is a multiple of $\operatorname{lcm}(12,14,15)=420.$

Purple candy costs $20$ cents, so $n=\dfrac{M}{20}.$ The smallest $M$ is $420,$ giving $n=\dfrac{420}{20}=21.$

Thus, B is the correct answer.

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