1967 AMC 12 Problem 24

Attempt Problem 24 of the 1967 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 1967 AMC 12 solutions, or check the answer key.

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24.

The number of solution-pairs in positive integers of the equation

3x+5y=501

is:

33

34

35

100

none of these

Reduce the equation modulo

5

to find the form of

x

Count the positive values

x=2+5k

that leave

y\gt0

Modulo

5,

the equation gives

3x\equiv1,

x\equiv2\pmod5.

Write

x=2+5k.

Then

y=\frac{501-3x}{5}=99-3k.

Positivity requires

k=0,1,\ldots,32,

giving

33

pairs.

Therefore, the correct answer is A.

Problem 24 in other years