AMC8 2005

AMC8 2005 · Q8

AMC8 2005 · Q8. It mainly tests Parity (odd/even).

Suppose m and n are positive odd integers. Which of the following must also be an odd integer?

假设 m 和 n 是正奇整数。以下哪一项一定是奇整数？

(A) m + 3n m + 3n

(B) 3m - n 3m − n

(D) (nm + 3)² (nm + 3)²

(E) 3mn 3mn

Answer

Correct choice: (E)

正确答案：（E）

Solution

(E) To check the possible answers, choose the easiest odd numbers for $m$ and $n$. If $m=n=1$, then $m+3n=4,\quad 3m-n=2,\quad 3m^2+3n^2=6,\quad (mn+3)^2=16\text{ and }3mn=3.$ This shows that (A), (B), (C) and (D) can be even when $m$ and $n$ are odd. On the other hand, because the product of odd integers is always odd, $3mn$ is always odd if $m$ and $n$ are odd. Questions: Which of the expressions are always even if $m$ and $n$ are odd? What are the possibilities if $m$ and $n$ are both even? If one is even and the other odd?

（E）为了检验可能的答案，给 $m$ 和 $n$ 选取最简单的奇数。若 $m=n=1$，则 $m+3n=4,\quad 3m-n=2,\quad 3m^2+3n^2=6,\quad (mn+3)^2=16\text{，且 }3mn=3.$ 这表明当 $m$ 和 $n$ 为奇数时，（A）、（B）、（C）和（D）可以是偶数。另一方面，因为奇整数的乘积总是奇数，所以如果 $m$ 和 $n$ 为奇数，则 $3mn$ 总是奇数。问题：当 $m$ 和 $n$ 为奇数时，哪些表达式总是偶数？若 $m$ 和 $n$ 都为偶数，会有哪些可能？若一个为偶数另一个为奇数，又有哪些可能？

Topics

NT.005 Parity (odd/even)