The positive integers \(N\) and \(N^2\) both end in the same sequence of four digits \(abcd\) when written in base 10, where digit a is not zero. Find the three-digit number \(abc\).

nchang Apr 1, 2020

#1**+1 **

I think we need to use modular arithmetic. This is how far I got \(n^2 - n = n(n - 1)\equiv 0\mod{10000}\).

I just realized that my "n" needs to be capitalized but whatever... :)

nchang Apr 1, 2020

#2**-1 **

Notice that the digits before \(abcd\) are not at all that important. Beyond that, I'm still thinking. Perhaps try modulo arithmetic? d has to be 1, 5, or 6.

Impasta Apr 1, 2020