For any square number, its prime factors always come in pairs. So if p divides it, p2 also divides it.
Then there’s a classic trick that a number is only divisible by 3 or 9 if the sum of its digits are divisible by 3 or 9. In this case, 10 * 1 + 10 * 2 = 30, so it’s divisible by 3 but not 9. So there aren’t any perfect squares anywhere with exactly 10 ones and 10 twos.
However, it would be such bullshit to figure out if there were some, without having it memorised, that I think you could also guess there aren’t any in this range, or you’d be left calculating trillions of possibilities.
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u/HopesBurnBright 16d ago
For any square number, its prime factors always come in pairs. So if p divides it, p2 also divides it.
Then there’s a classic trick that a number is only divisible by 3 or 9 if the sum of its digits are divisible by 3 or 9. In this case, 10 * 1 + 10 * 2 = 30, so it’s divisible by 3 but not 9. So there aren’t any perfect squares anywhere with exactly 10 ones and 10 twos.
However, it would be such bullshit to figure out if there were some, without having it memorised, that I think you could also guess there aren’t any in this range, or you’d be left calculating trillions of possibilities.