Without computing them, find out which of the following arithmetic expressions are even: 43 + 37, 672 - 348, × 3, 708 - 477, 809 + 214, , 543 - 479, 513³
Parity Rules: odd + odd = even even + even = even odd + even = odd even - even = even odd - odd = even odd - even = odd even - odd = odd For multiplication: any expression with an even factor is even; odd × odd = odd
43 + 37: 43 is odd, 37 is odd odd + odd = even ∴ 43 + 37 is even
672 - 348: 672 is even, 348 is even even - even = even ∴ 672 - 348 is even
× 3: Contains factor 4 (even) Any expression with an even factor is even ∴ × 3 is even
708 - 477: 708 is even, 477 is odd even - odd = odd ∴ 708 - 477 is odd (NOT even)
809 + 214: 809 is odd, 214 is even odd + even = odd ∴ 809 + 214 is odd (NOT even)
: 119 is odd, 303 is odd odd × odd = odd ∴ is odd (NOT even)
543 - 479: 543 is odd, 479 is odd odd - odd = even ∴ 543 - 479 is even
513³: 513 is odd odd³ = odd ∴ 513³ is odd (NOT even)
Answer: The even expressions are 43 + 37, 672 - 348, × 3, and 543 - 479.
Explanation
The chapter explains parity rules: odd + odd = even, even + even = even, odd + even = odd, even - even = even, odd - odd = even, odd - even = odd, even - odd = odd. For multiplication: any expression with an even factor is even; odd × odd = odd. Applying these: 43+37 (odd+odd=even), 672-348 (even-even=even), ×3 (contains 4, so even), 708-477 (even-odd=odd), 809+214 (odd+even=odd), (odd×odd=odd), 543-479 (odd-odd=even), 513³ (odd³=odd).
Solution Steps
Step 1: Apply parity rules for addition/subtraction
Step 2: odd + odd = even: 43 + 37 is even
Step 3: even - even = even: 672 - 348 is even
Step 4: Expression with factor 4 is even: × 3 is even
Step 5: even - odd = odd: 708 - 477 is odd
Step 6: odd + even = odd: 809 + 214 is odd
Step 7: odd × odd = odd: is odd
Step 8: odd - odd = even: 543 - 479 is even
Step 9: odd³ = odd: 513³ is odd