명제의 증명과 절대부등식 연습문제 — Sundstrom, Mathematical Reasoning · 3.3절 · 2d번

Sundstrom, Mathematical Reasoning · 3.3절 · 2d번영어 원문

Are the following statements true or false? Justify each conclusion.

For all integers $a$ and $b$ , if $a$ is odd and $b$ is odd, then $4$ divides $\left( a^2 + 3b^2 \right)$ .

정답·풀이 보기 (영어 원문)

This statement is true. Use a direct proof. Let $a$ and $b$ be integers and assume they are odd. So there exist integers $m$ and $n$ such that

$a = 2m + 1 \qquad \text{ and } \qquad b = 2n + 1$

. We then see that

$\begin{aligned}a^2 + 3b^2 & = 4m^2 + 4m + 1 + 12n^2 + 12n + 3 \\ & = 4\left( m^2 + m + 3n^2 + 3n + 1 \right)\end{aligned}$

. This shows that 4 divides $a^2 + 3b^2$ .

한 번 더 — 비슷한 문제Sundstrom, 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 $a \mid \left(b-c \right)$ .

다 풀었으면 정답 확인

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 $b - c$ .