Question

Prove that $a|b$ if and only if $a^n|b^n$.

Short answer

It is proved that $a|b$ if and only if $a^n|b^n$.

Step-by-step solution

Obtain the a|b if and only if an|bn 

Consider the equation $a|b$ if and only if $a^n|b^n$.

If $a|b\Rightarrow b=at$ for some integer of t , this gives role="math" localid="1646219985780" $b^n=atat=a^n{t}^n$ . Thus it proves that $a^n{b}^n$.

Find proof

Now consider that $a^n|b^n$.

Let’s assume that,

$a={p_1}^{r_1}{p_2}^{r_2}...{p_k}^{r_k}$and $b={p_1}^{s_1}{p_2}^{s_2}...{p_k}^{s_k}$. Here,$r_i,s_i\ge 0$ .

$a^n|b^n$gives ${p_1}^{n{r}_1}{p_2}^{n{r}_2}...{p_k}^{n{r}_k}|{p_1}^{n{s}_1}{p_2}^{n{s}_2}...{p_k}^{n{s}_k}$ . This indicates that $n{r}_i\le n{s}_i$.

Here, i gives $r_i\le s_i$.

Therefore,

If $a|b\Rightarrow b=at$ for some integer of t , it gives $b^n=atat=a^n{t}^n$ . This proves that $a^n|b^n$.

Hence, it is proved that $a|b$if and only if $a^n|b^n$.