2019 AMC 12B Problem 5

Problem 5 of 25EasierNumber Theory

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?

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Solution

Let

M

be Casper’s money in cents. Since he can exactly buy

12,

14,

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

420,

giving

n=\dfrac{420}{20}=21.

Thus, B is the correct answer.