1963 AMC 12 Problem 31

Attempt Problem 31 of the 1963 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 1963 AMC 12 solutions, or check the answer key.

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31.

The number of solutions in positive integers of

2x+3y=763

is:

255

254

128

127

0

Answer: D

Concepts:Diophantine Equation parity counting integers in a range

Difficulty rating: 1710

Small hint:

Solve for

x=\frac{763-3y}{2}

Big hint:

Positive integral

x

requires

y

to be an odd positive integer below

\frac{763}{3}

Solution:

We need

y

odd so that

763-3y

is even, and positivity gives

1\leq y\leq253.

The odd values

1,

3,

\ldots,

and

253

number

\frac{253+1}{2}=127.

Therefore, the correct answer is D.