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
Expressions Always Even:
2a + 2b: 2a + 2b = 2(a + b) Always has factor 2 ∴ Always even
4m + 2n: 4m + 2n = 2(2m + n) Always has factor 2 ∴ Always even
2u - 4v: 2u - 4v = 2(u - 2v) Always has factor 2 ∴ Always even
13k - 5k: 13k - 5k = 8k Always has factor 8 ∴ Always even
4k × 3j: 4k × 3j = 12kj Always has factor 4 ∴ Always even
Expressions Not Always Even:
3g + 5h: odd + odd = even, even + even = even, but odd + even = odd ∴ Not always even
6m - 3n: 6m is even, but 3n can be odd or even 6m - 3n = 3(2m - n) If m = 1, n = 1, result is 3 (odd) ∴ Not always even
x² + 2: If x is odd, x² is odd, so x² + 2 is odd If x = 3, x² + 2 = 11 (odd) ∴ Not always even
b² + 1: If b is even, b² is even, so b² + 1 is odd If b = 2, b² + 1 = 5 (odd) ∴ Not always even
Explanation
The chapter explains analyzing parity algebraically. 2a + 2b = 2(a+b) always has factor 2, so always even. 3g + 5h: if g and h are both odd or both even, result is even; if one odd one even, result is odd — not always even. 4m + 2n = 2(2m+n) always has factor 2. 2u - 4v = 2(u-2v) always has factor 2. 13k - 5k = 8k always has factor 8. 6m - 3n = 3(2m-n): if m=1, n=1, result is 3 (odd). x² + 2: if x=3, x²+2=11 (odd). b² + 1: if b=2, b²+1=5 (odd). 4k × 3j = 12kj always has factor 4.
Solution Steps
Identify expressions with a common factor of 2 (or multiple of 2) in all terms.
Factor each expression to check for even factors.
For expressions without a guaranteed even factor, test with specific integer values.
Classify each expression as 'always even' or 'not always even' based on the analysis.