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3 Responses
When I do a “could this be a square?” triage, I run a few safe filters that never throw out a true square, like a friendly bouncer checking IDs before letting you into the math club: (1) Last digit must be 0, 1, 4, 5, 6, or 9 (because squares mod 10 are exactly those). (2) Digital root must be 0, 1, 4, or 7 (equivalently, n ≡ 0,1,4,7 mod 9), since squares mod 9 are exactly 0,1,4,7. (3) Check mod 8 using the last three decimal digits: squares are 0, 1, or 4 mod 8; in particular, every odd square is 1 mod 8. (4) Quick last-two-digit refinements from squares mod 100: if it ends in 0, it must end in 00; if it ends in 5, it must end in 25; if it ends in 6, the tens digit must be odd; if it ends in 4, the tens digit must be even; if it ends in 1 or 9, the tens digit must be even. Each of these is necessary (never excludes a real square), and together they rule out most numbers very fast-for example, 123456 ends with 6 and has odd tens (so it slips past rule 4), but its digital root is 3 (not in 0,1,4,7), so out it goes. I sometimes worry I’m overthinking it, but this little sieve usually catches the impostors before I even think about a square root.
Quick, safe filters (none will ever toss a true square): last digit must be 0,1,4,5,6,9 (and if it ends in 5 it must be 25; if it ends in 0 it must be 00); digital root must be 1,4,7,9 (i.e., ≡0,1,4,7 mod 9); and n ≡ 0 or 1 (mod 4) which you can check from the last two digits. I’m pretty sure that’s the cleanest mental set-add n ≡ 0,1,4 (mod 8) via the last three digits if you want extra bite; background: https://en.wikipedia.org/wiki/Square_number.
My quick “bouncer” checks are: last digit must be 0,1,4,5,6,9-and if it’s 0 it must end with 00, if it’s 5 it must end with 25 (squares have even powers of 2 and 5); the digital root must be 1,4,7, or 9 (squares ≡ 0,1,4,7 mod 9); and using just the last four digits (since 10000 is a multiple of 16) you need n ≡ 0,1,4, or 9 mod 16.
I’m pretty sure this compact trio never tosses a real square and knocks out most pretenders-like a friendly airport security line for numbers!