2004 AMC 12A Problem 3 — Solution | LIVE by Po-Shen Loh

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2004 AMC 12A Problem 3

Attempt Problem 3 of the 2004 AMC 12A 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 2004 AMC 12A solutions, or check the answer key.

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

3.

For how many ordered pairs of positive integers $(x, y)$ is $x + 2y = 100?$

$33$

$49$

$50$

$99$

$100$

Answer: B

Concepts:Diophantine Equation counting integers in a range

Difficulty rating: 1080

Solution:

Writing $x = 100 - 2y,$ the value of $x$ is a positive integer precisely when $y$ is a positive integer with $1 \le y \le 49.$

This gives $49$ valid ordered pairs.

Thus, the correct answer is B.

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