Repdigit endings to squares | Colin

Over at @onthisdayinmath, Pat highlights a @jamestanton question about squares:

$2^2$ ends with 4 and $12^2$ ends with 44. Is there a square than ends 444? How about one that ends 4444?

Pat’s answer (yes to the first — $38^2 = 1444$ is the smallest — and probably not to the second) is correct, but I wanted to dig into why no square ends 4444.

One bit of scaffolding I’m going to call on slightly later (a lemma, if you want a technical term to impress people with) is that no square number ends in 11. The quickest way to prove this is to think modulo 20: the only possible remainders…

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