Is the expression even or odd?

The idea

An algebraic expression can take values of different parity for different values of its letter-number, while some expressions always have even parity whatever value the letter-number takes.

One expression, many numbers

Take the expression 3n + 4. When n = 3, it is 3 × 3 + 4 = 13, which is odd.

What happens when n = 8?

• It is odd again
• It is even
• We cannot say without calculating

Slide the letter-number

Slide *n* and watch the value. For each expression, try at least four values. Then decide: is it always even, always odd, or sometimes each?

It can change, or never change

The parity of 3n + 4 depends on *n*:

But some expressions are always even, like 100p and 48w − 2. They are made of whole lots of pairs. And 6k + 2 is always even, yet it gives only 8, 14, 20, … for k = 1, 2, 3, …: many even numbers are missing.

An algebraic expression can take values of different parity for different values of its letter-number, while some expressions always have even parity whatever value the letter-number takes.

• n | 3n + 4 | Parity
• 3 | 13 | odd
• 8 | 28 | even
• 10 | 34 | even

Notes

An algebraic expression can take values of different parity for different values of its letter-number, while some expressions always have even parity whatever value the letter-number takes.

Check yourself

Which expression is always even, whatever the counting number *n* is?

For which value of *n* is 3n + 4 even?

Priya says: “6k + 2 is always even, so it gives every even number.” Which even number does it miss?

Find the value of 48w − 2 when w = 3.

Answer: 142

48 × 3 − 2 = 144 − 2 = 142. It is even, as 48w − 2 always is.

• 20n — correct. Yes! 20n is n lots of 20, and 20 is even, so it is always in pairs.
• 20n + 1. 20n is even, but the +1 leaves a dot over every time: it is always odd.
• n + 20. 20 is even, but n can be odd or even, so n + 20 can be either.
• 3n + 20. 3n has the parity of n, so 3n + 20 is odd for odd n and even for even n.
• n = 5. 3 × 5 + 4 = 19, which is odd.
• n = 7. 3 × 7 + 4 = 25, which is odd.
• n = 4 — correct. Yes! 3 × 4 + 4 = 16, which is even.
• n = 9. 3 × 9 + 4 = 31, which is odd.
• 8. 8 is there: k = 1 gives 6 × 1 + 2 = 8.
• 10 — correct. Yes! 6k + 2 gives 8, 14, 20, … and 10 is not there: no counting number k gives 6k + 2 = 10.
• 14. 14 is there: k = 2 gives 6 × 2 + 2 = 14.
• 20. 20 is there: k = 3 gives 6 × 3 + 2 = 20.