2025 AMC 12B Problem 5

Attempt Problem 5 of the 2025 AMC 12B 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 2025 AMC 12B solutions, or check the answer key.

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

Positive integers $x$ and $y$ satisfy the equation $57x + 22y = 400.$ What is the least possible value of $x + y?$

$10$

$11$

$13$

$14$

$15$

Answer: E

Concepts:Diophantine Equation modular arithmetic

Difficulty rating: 1290

Solution:

Modulo $22,$ the equation gives $13x \equiv 4,$ so $x \equiv 2 \pmod{22}.$ With $57x \lt 400,$ the only option is $x = 2,$ which gives $22y = 286,$ so $y = 13.$ Then $x + y = 15.$

Thus, the correct answer is E.

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