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.
Expressions that always give even numbers: 2a + 2b (both terms even) 4m + 2n (both terms even) 2u - 4v (both terms even) 13k - 5k = 8k (even) 4k × 3j (has factor 4)
Expressions that may not always be even: 3g + 5h (odd + odd = even, but odd + even = odd) 6m - 3n (even - odd = odd) x² + 2 (even² + 2 = even, odd² + 2 = odd) b² + 1 (even² + 1 = odd, odd² + 1 = even)
Explanation
An expression always gives an even number if it has a factor of 2 for all integer values. 2a + 2b = 2(a+b) always even. 4m + 2n = 2(2m+n) always even. 2u - 4v = 2(u-2v) always even. 13k - 5k = 8k always even. 4k × 3j = 12kj always even. For 3g + 5h: if g and h are both odd, result is even; if one is even, result is odd. For x² + 2: if x is even, x² is even, so x² + 2 is even; if x is odd, x² is odd, so x² + 2 is odd.
Solution Steps
2a + 2b: both terms even
4m + 2n: both terms even
2u - 4v: both terms even
13k - 5k = 8k: even
4k × 3j: has factor 4
3g + 5h: odd + odd = even, but odd + even = odd
6m - 3n: even - odd = odd
x² + 2: even² + 2 = even, odd² + 2 = odd
b² + 1: even² + 1 = odd, odd² + 1 = even