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Number theory Difficulty 4.7 AIME Prove it South Africa

Are there infinitely many integers whose square ends in three 4s, i.e. ...444?

Solution

Notice that $38^2 = 1444$ which ends in ...444.

Moreover, $(1000k + 38)^2 = 1000000k^2 + 76000k + 1444$ which ends in ...444 for any natural number $k$ . Thus there are infinitely many such numbers.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.