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### Odd Digits of Square Numbers

```
Date: 02/07/99 at 12:42:55
From: Kevin Bryceland
Subject: Odd square numbers

Dear Dr. Math,

I cannot work out why there are no square numbers other than 1 and 9

Yours thankfully,
Kevin Bryceland
```

```
Date: 02/08/99 at 17:30:48
From: Doctor Nick
Subject: Re: Odd square numbers

Hi Kevin -

The reason that these are the only two squares that have only odd
digits is that a square number with more than 1 digit will have at
least one even digit in the first two (that is, the right two) places.

If n is even, then the right-most digit of n^2 is even.

If n is odd, then the right-most digit of n^2 is odd, but the next
digit is always even!

A simple way to prove this is to use the fact that the right two digits
of n^2 depend entirely on the right two digits of n. That means that we
only have to check the squares of 1, 3, 5, 7, 9, 11, ..., 99 to see
that they have an even number in the second place. You can cut the
effort in half by noting that 1^2 and 99^2 end in the same two digits,
3^2 and 97^2 end in the same two digits, and so on, so you really only
need to check the squares of 1, 3, 5, 7, ..., 49, which isn't too bad.

In fact, you can cut it down even further. Notice that

(50-m)^2 = 50^2 - 100m + m^2

so if 0 < m < 50, then m^2 and (50-m)^2 end in the same two digits.
That means you only have to check 1, 3, 5, 7, ..., 25.

Notice that the fact that we're using base 10 is very important here.
In other bases there could be squares that have only odd digits.

Have fun,

- Doctor Nick, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 04/01/01 at 13:17:04
From: Doctor Schwa
Subject: Re: Odd square numbers

To continue Dr. Nick's excellent response, you can cut down the checking
even more by looking at

(10a + b)^2 = 100a^2 + 20ab + b^2

The first term, 100a^2, doesn't affect the tens place at all; 20ab only
changes the tens place by an even number, so you only need to check the
squares of the one-digit numbers to verify that the tens place or the ones
place is always even.

As Dr. Nick already pointed out, if the ones place (b) is even, then b^2
will end with an even digit as well.

So now there are only five numbers that need to be checked.

--Dr. Schwa
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory
Middle School Number Sense/About Numbers

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