How many square numbers have no even digits?

The answer is: two (1 and 9).

Bolaji Atanda won the prize. He experimented with a calculator, and concluded:

a. there are no other square numbers with no even digits; and, even more:

b. in every square number one of the last two digits, the tens digit or the units digit, is even.

**Proof**

You can see by looking that this is true for square numbers up to 100.

Every square number bigger than 100 is (10m+n)^{2} for some m and some n<10

(10m+n)^{2} = 100m^{2} + 20mn + n^{2}

so the last-but-one digit of the square number is even (because the last digit of 2mn is even) unless the tens digit of n^{2} is odd.

But 5, 7, and 9 are the only values of n which can give us an odd last digit of the square number.

5^{2}=25, 7^{2}=49, 9^{2}=81. In each case the tens digit is even.

Therefore either the last digit, or the last-but-one digit, of every square number is even. ▇

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