题目

Problem

(i) A student states

“If $x$ and $y$ are irrational numbers, $x\ne y$ , then $xy$ is also irrational.”

Show, by counter example, that this statement is not always true.

(1)

(ii) Prove, using algebra, that for all odd integers $n$ , the value of the expression

n^3+3n+2

is always even but never a multiple of $4$

(4)

解答

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反例只需要找一组不同的无理数，使乘积变成有理数。常用想法是用根号数相乘。

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Let

\begin{align*} x=\sqrt2,\qquad y=\sqrt8. \end{align*}

Then $x$ and $y$ are irrational and $x\ne y$ .

But

\begin{align*} xy=&\,\sqrt2\sqrt8\\ =&\,\sqrt{16}\\ =&\,4, \end{align*}

which is rational.

Therefore, the statement is not always true.

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任意奇数都可写成 $2k+1$ 。代入后要说明结果是 $4$ 的倍数再加 $2$ ，所以一定是偶数，但不能被 $4$ 整除。

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Let

\begin{align*} n=2k+1, \end{align*}

where $k$ is an integer.

Then

\begin{align*} n^3+3n+2 =&\,(2k+1)^3+3(2k+1)+2\\ =&\,(8k^3+12k^2+6k+1)+6k+3+2\\ =&\,8k^3+12k^2+12k+6\\ =&\,4(2k^3+3k^2+3k+1)+2. \end{align*}

This expression is $4$ times an integer plus $2$ .

Therefore it is even, but it leaves a remainder of $2$ when divided by $4$ , so it is never a multiple of $4$ .