AMC 8 · 2005 · #8
Grade 6 arithmeticPick an answer.
AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The actual values of m and n are not given — only their parity. Tool #12 (Use Parity or Modular Arithmetic) is built for exactly this: track whether each piece is odd or even using two short rules (odd ± odd = even, odd × odd = odd) and ignore the numerical size. Once the parity of each choice is fixed, Tool #3 (Eliminate Possibilities) sweeps away any choice whose parity comes out even. Only one survives.
Two parity rules settle every choice: odd ± odd = even, and odd × odd = odd. With m, n odd, 3m and 3n are odd too.
These are the only two parity facts the problem needs — everything else is rule-chasing.
2.OA.C.3Draw A Venn Diagram(A) m + 3n is odd + odd, so it is always even — eliminate it.
Two odds combine to an even — same way two odd handfuls of coins pair up perfectly.
2.OA.C.3Eliminate Possibilities(B) 3m - n is odd - odd, which is even — eliminate.
Subtraction obeys the same parity rule as addition.
2.OA.C.3Eliminate Possibilities(C) 3m² + 3n²: each term is odd, and odd + odd is even — eliminate.
The exponent and the leading 3 do not change parity — both pieces stay odd, and the sum is still even.
3.OA.B.5Eliminate Possibilities(D) (nm + 3)²: inside is odd + odd = even, and even squared is even — eliminate.
Squaring never rescues parity — even squared is still even.
3.OA.B.5Eliminate Possibilities(E) 3mn is odd × odd × odd = odd — the only choice that must be odd.
A product of odd numbers stays odd no matter how many you multiply — so (E) is forced.
6.EE.A.2Draw A Venn DiagramWhen a problem only tells you the parity of the inputs, throw away the actual numbers and track "odd" or "even" through each step. Two short rules (odd ± odd = even, odd × odd = odd) decide all five choices in this AMC 8 problem at once.