We know how to identify even numbers. Without computing them, find out which of the following arithmetic expressions are even. 43 + 37, 672 – 348, 4 × 347 × 3, 708 – 477, 809 + 214, 119 × 303, 543 – 479, 513³
Answer
Four of them are even: 43 + 37, 672 – 348, 4 × 347 × 3 and 543 – 479.
Expression
Parity reasoning
Even?
43 + 37
odd + odd = even
Yes
672 – 348
even – even = even
Yes
4 × 347 × 3
a product with the factor 4 in it must be even
Yes
708 – 477
even – odd = odd
No
809 + 214
odd + even = odd
No
119 × 303
odd × odd = odd
No
543 – 479
odd – odd = even
Yes
513³
odd × odd × odd = odd
No
Check it yourself (afterwards!): 80, 324, 4164, 231, 1023, 36057, 64 and 135005697. The parity rules got every one right without any of that work.
Q3.
Using our understanding of how parity behaves under different operations, identify which of the following algebraic expressions give an even number for any integer values for the letter-numbers. 2a + 2b, 3g + 5h, 4m + 2n, 2u – 4v, 13k – 5k, 6m – 3n, x² + 2, b² + 1, 4k × 3j
Answer
Five of the nine are always even: 2a + 2b, 4m + 2n, 2u – 4v, 13k – 5k and 4k × 3j. Each of them has 2 as a factor of the whole expression.
Expression
Rewritten
Always even?
2a + 2b
2(a + b)
Yes
3g + 5h
—
No
4m + 2n
2(2m + n)
Yes
2u – 4v
2(u – 2v)
Yes
13k – 5k
8k = 2(4k)
Yes
6m – 3n
3(2m – n)
No
x² + 2
—
No
b² + 1
—
No
4k × 3j
12kj = 2(6kj)
Yes
Why it happens: once an expression is written as 2 × (something whole), 2 is a factor of it no matter what the letters stand for. The four that fail have no such factor — a single counterexample kills each one: 3g + 5h = 3 when g = 1, h = 0; 6m – 3n = 3 when m = n = 1; x² + 2 = 11 when x = 3; b² + 1 = 5 when b = 2.