2025 AMC 12B Problem 5

Problem 5 of 25EasierNumber Theory

Positive integers

x

and

y

satisfy the equation

57x + 22y = 400.

What is the least possible value of

x + y?

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Solution

Modulo

22,

the equation gives

13x \equiv 4,

x \equiv 2 \pmod{22}.

With

57x \lt 400,

the only option is

x = 2,

which gives

22y = 286,

y = 13.

Then

x + y = 15.

Thus, the correct answer is E.

Concepts: Diophantine Equation · modular arithmetic

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.