Unique 2

Number Theory Level 3

A natural number $m$ has 4 digits.

$m^2$ ends with the same 4 digits of $m$

Find product of the digits of $m$

1 solution

Mr. India
May 13, 2019

$m^2-m=m(m-1)$ is divisible by $10^4$

Now, one out of $m$ and $m-1$ is odd and other is even.

factoring $10^4=625×16$

Taking $m$ as even part we have $m\equiv 0\pmod{16}$ and $m-1\equiv 0\pmod{625}$

This gives the solution as $9376$

Taking $m$ to be odd gives no solution in 4 digits

But why can we take m as the even part?

Jesse Li - 2 years ago