2019 AMC 10B Problem 7

Below is the professionally curated solution for Problem 7 of the 2019 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 10B solutions, or check the answer key.

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Concepts:least common multiple divisibility

Difficulty rating: 1140

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 the number of cents he has $c.$ Then, $c$ is a multiple of $12,14,$ and $15.$ Thus, it must be a multiple of $420.$

Let $c=420k$ for some $k.$ Also, $c=20n,$ so $420k=20n,$ making $n=21k.$ Since $k$ is a whole number, the minimum possible value of $n$ is $21.$

Thus, the answer is B .

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