1980 AMC 12 Problem 15

Attempt Problem 15 of the 1980 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1980 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

15.

A store prices an item in dollars and cents so that when

4\%

sales tax is added no rounding is necessary because the result is exactly

n

dollars, where

n

is a positive integer. The smallest value of

n

1

13

25

26

100

Let the untaxed price be an integer number

p

of cents

Reduce the equation

1.04p=100n

to a divisibility condition on

n

Let the price be

p

cents. Then

\frac{104}{100}p=100n,

13p=1250n.

Since

13

and

1250

are relatively prime,

13

must divide

n.

The smallest positive possibility is

n=13.

Therefore, the correct answer is B.

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