2019 AMC 10B Problem 7

Problem 7 of 25EasierNumber Theory

Each piece of candy in a shop 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 least possible value of

n?

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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,

420k=20n,

making

n=21k.

Since

k

is a whole number, the minimum possible value of

n

21.

Thus, the answer is B .

Tagged: least common multiple · divisibility