명제의 증명과 절대부등식 연습문제 — Sundstrom, Mathematical Reasoning · 3.1절 · 1a번

Sundstrom, Mathematical Reasoning · 3.1절 · 1a번영어 원문

Prove each of the following statements:

For all integers $a$ , $b$ , and $c$ with $a \ne 0$ , if $a \mid b$ and $a \mid c$ , then $a \mid \left(b-c \right)$ .

정답·풀이 보기 (영어 원문)

Since $a \mid b$ and $a \mid c$ , there exist integers $m$ and $n$ such that $b = am$ and $c = an$ . Hence,

$\begin{aligned}b - c & = am - an \\ & = a(m - n)\end{aligned}$

Since $m - n$ is an integer (by the closure properties of the integers), the last equation implies that $a$ divides $b - c$ .

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 3.1절 · 4a번영어 원문

Complete the following.

If $x$ and $y$ are integers and $xy=1$ , explain why $x=1$ or $x=-1$ .

다 풀었으면 정답 확인

If $xy = 1$ , then $x$ and $y$ are both divisors of $1$ , and the only divisors of $1$ are $-1$ and $1$ .