Square Kou and his Not-So-Square Uncle

Number Theory Level pending

A 3 digit number have its square made up by just 3 non-zero distinct digits. Kouhan told his Uncle Ren(zoku) about the number and its property to impress him, but to his surprise, Uncle Ren one-up him by multiplying that square to a 2 digit number and have the answer to have the exact same digits (and the number of those digits too) as they were in Kouhan's square, plus 2 zeros.

What is the palindromic difference between the 3-digit and 2-digit numbers?

Notes : As there are multiple possible answers, the intended one read the same from the back. Special thanks to Paul Hindess for his help and advice in making a better question in every aspect (mathematical and the language part, too).

The answer is 676.

1 solution

Ricardo Moritz Cavalcanti
Dec 22, 2020

I wrote a program in Python to do a brute force search and found only three combinations of 3- and 2-digit numbers satisfying the conditions of the problem:

(1) $m = 111,\, n = 82\, (m^2 = 12321,\, nm^2 = 1010322$ );

(2) $m = 688,\, n = 91\, (m^2 = 473344,\, nm^2 = 43074304$ );

(3) $m = 765,\, n = 89\, (m^2 = 585225,\, nm^2 = 52085025$ ).

Only in the third case the difference $m - n\, (= 676)$ is a palindromic number.

Square Kou and his Not-So-Square Uncle

The answer is 676.

1 solution

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