riddle of the day

Some numbers leave a signature when multiplied by themselves. They hide at the end of their own square, as if the calculation returned them intact.

The Square That Ends in Itself

Strategist
Numerical territory

Riddle statement

Find a two-digit number with this property:

when you square it, the last two digits of the result are again the number itself.

For example:

25^2 = 625

which ends in 25.

Is there another two-digit number with this property?

Show solution

Solution

The only two-digit numbers are 25 and 76.

The condition that the square ends in the number itself is:

$ n^2\equiv n\pmod{100}, $

that is:

$ n(n-1)\equiv0\pmod{100}. $

Since 100 is $4\cdot25$, solve separately modulo 4 and modulo 25. Two consecutive integers are coprime, so for their product to be divisible by a prime power, all of that prime power must divide one of the two factors.

Therefore:

$ n\equiv0\text{ or }1\pmod4 $

and

$ n\equiv0\text{ or }1\pmod{25}. $

Combining the four possibilities gives, modulo 100:

$ 0,\ 1,\ 25,\ 76. $

The first two are not two-digit numbers. The remaining ones are:

$ 25^2=625 $

and

$ 76^2=5776. $

Answer: the only two-digit numbers with this property are 25 and 76.