2024 AMC 12A Problem 9

Problem 9 of 25EasierAlgebra Number Theory

Let

M

be the greatest integer such that both

M+1213

and

M+3773

are perfect squares. What is the units digit of

M?

Show solution

Solution

Write

M+1213=a^2

and

M+3773=b^2,

b^2-a^2=2560,

i.e.

(b-a)(b+a)=2560.

Both factors have the same parity, hence both even. To maximize

a

(and thus

M

), minimize

b-a:

take

b-a=2,\ b+a=1280,

a=639.

Then

M=639^2-1213

=408321-1213

=407108,

whose units digit is

8.

Thus, the correct answer is E.