Always even?

The idea

An algebraic expression gives an even number for all integer values of its letter-numbers if every term is even, or if it can be rewritten as 2 times another expression, so that 2 is a factor.

A claim to test

A friend says: “The expression 4m + 2n is an even number, whatever whole numbers m and n stand for.” Try m = 3 and n = 2: 4 × 3 + 2 × 2 = 16. Even!

Is your friend right for every whole number m and n, positive or negative?

• Yes, always even
• Only for positive numbers
• No, sometimes odd

Even every time?

For each expression, change the letters and watch the result. Press Rewrite it to see whether a 2 can come out. Then say Always even or Not always.

Two ways to be sure

An algebraic expression gives an even number for all integer values of its letter-numbers if every term is even, or if it can be rewritten as 2 times another expression, so that 2 is a factor.

4m + 2n: 4m is even and 2n is even, so the sum is even. Or: 4m + 2n = 2(2m + n), and 2 is a factor.

Careful: 13k − 5k has odd terms for odd k, but it simplifies to 8k = 2(4k). Simplify first, then look for the 2.

To show an expression is not always even, one example that gives an odd number is enough.

Notes

An algebraic expression gives an even number for all integer values of its letter-numbers if every term is even, or if it can be rewritten as 2 times another expression, so that 2 is a factor.

Check yourself

Which expression is always even, for any whole numbers a and b?

10p + 6q = 2(5p + ? q). What number replaces the ?

Answer: 3

6q = 2 × 3q, so 10p + 6q = 2(5p + 3q).

Is 7k − 3k always even, for any whole number k?

Which values show that 3g + 5h is not always even?

• 5a + 3b. Try a = 1 and b = 0: the value is 5, which is odd.
• 7a − 4b. Try a = 1 and b = 0: the value is 7, which is odd.
• 6a − 8b — correct. Yes! 6a − 8b = 2(3a − 4b), so 2 is a factor.
• 2a + 3b. Try a = 0 and b = 1: the value is 3, which is odd.
• Yes, because 7k − 3k = 4k = 2(2k) — correct. Yes! Simplify first: 7k − 3k = 4k, and 2 is a factor of 4k.
• No, because 7k is odd when k is odd. True, 7k is odd when k is odd. But 3k is odd then too, and odd − odd = even. In fact 7k − 3k = 4k.
• No, because 7 and 3 are odd numbers. The terms alone do not decide it. Simplify: 7k − 3k = 4k = 2(2k), always even.
• g = 2 and h = 4. That gives 6 + 20 = 26, which is even. An example that gives an even number proves nothing.
• g = 1 and h = 1. That gives 3 + 5 = 8, which is even. An example that gives an even number proves nothing.
• g = 2 and h = 1 — correct. Yes! 3 × 2 + 5 × 1 = 11, an odd number. One such example is enough.
• g = 3 and h = 3. That gives 9 + 15 = 24, which is even. An example that gives an even number proves nothing.