Parity of a square

The idea

The square x² of a number x has the same parity as x: x² is even if x is even and odd if x is odd.

A square of dots

Put dots in a square, 5 dots on each side. That is 5 × 5 = 25 dots. Now try to pair them up, two by two.

What happens to the dots?

• All of them pair up
• Exactly one dot is left over
• We cannot tell

Pair up the dots

Change the side x. Every pair of dots is joined. Look at the squares with an even side and with an odd side. Is a dot ever left over?

x² has the same parity as x

The square x² of a number x has the same parity as x: x² is even if x is even and odd if x is odd.

x even: every row pairs up, so nothing is left. x odd: each row leaves one dot, and those x dots are odd in number, so one dot stays unpaired.

So x² + 2 is not always even: for x = 6 it is 38, for x = 3 it is 11.

Notes

The square x² of a number x has the same parity as x: x² is even if x is even and odd if x is odd.

Check yourself

n² = 169. Without finding n: is n even or odd?

How many of 3², 8², 11², 14² and 20² are even numbers?

Answer: 3

A square has the parity of its number. 8, 14 and 20 are even, so 8², 14² and 20² are even (64, 196, 400). That is 3.

Is x² + 2 always an even number?

Is x² + x always even, whatever whole number x is?

• Even. If n were even, n² would be even. 169 is odd, so n is odd (n = 13).
• Odd — correct. Yes! A square has the same parity as the number, so n² = 169 odd means n is odd. (13 × 13 = 169.)
• We cannot tell. We can: x² and x always have the same parity. 169 is odd, so n is odd.
• Yes, because 2 is even. x² can be odd. For x = 3: 3² + 2 = 11, which is odd.
• No: it is odd when x is odd — correct. Yes! x² + 2 has the parity of x². For x = 6 it is 38 (even), for x = 3 it is 11 (odd).
• No, it is always odd. For x = 6, x² + 2 = 38, which is even.
• Yes: x² and x have the same parity, so their sum is even — correct. Yes! Two even numbers add to even, two odd numbers add to even. For x = 7: 49 + 7 = 56.
• No: it is odd when x is odd. When x is odd, x² is odd too, and odd + odd = even. (7² + 7 = 56.)
• No: x² is even only when x is even. True, but when x is odd, x² is odd and x is odd, and their sum is even again.