Mathematical Reflections 2012, Issue 4 - Problem J235

Problem:
In the equality $\sqrt{ABCDEF}=DEF$, different letters represent different digits. Find the six-digit number $ABCDEF$.

Proposed by Titu Andreescu.

Solution:
From the given equality, it must be $ABCDEF^2-DEF \equiv 0 \pmod {1000}$, and since $ABCDEF \equiv DEF \pmod{1000}$, we have $DEF^2-DEF=DEF(DEF-1) \equiv 0 \pmod{1000}$. Since $DEF$ and $DEF-1$ are coprime and $1000=2^3\cdot5^3$, one between this two numbers must be odd and divisibile by $5^3$ and the other must be divisible by $2^3$. Since $DEF$ and $DEF-1$ are three digit numbers, $DEF \in \{125,375,625,875\}$ or $DEF-1 \in \{125,375,625,875\}$. In the first case, $DEF-1$ is divisible by $8$ if and only if $DEF=625$, in the second case $DEF$ is divisible by $8$ if and only if $DEF-1=375$. So, $DEF=625,376$, but
$$625^2=390625 \qquad 376^2=141376,$$ i.e. the only number which satisfies the required conditions is $625$.