Question

Prove that $b|a$ if and only if$\left(-b\right)|a$ .

Short answer

It is proved that $b|a$ if and only if $\left(-b\right)|a$

Step-by-step solution

Prove that if b|a  then  -b|a

As $b|a$ then by definition of divisibility, we have $a=bx$ for some integer $x$.

Rewrite the condition $a=bx$ as:

localid="1646117797989" $a=\left(-b\right)\left(-x\right)$

This implies that $\left(-b\right)|a$. Hence if $b|a$then $\left(-b\right)|a$.

Prove that if -b|a  then  b|a

As $-b|a$ then by definition of divisibility, we have $a=\left(-b\right)\left(-x\right)$ for some integer $x$.

Rewrite the condition $a=\left(-b\right)\left(-x\right)$ as:

$a=bx$

This implies that $b|a$. Hence if $\left(-b|\right)a$then $b|a$.

Thus, it is proved that $b|a$ if and only if

localid="1652787199461" $\left(-b\right)|a$