Question

Prove that every integer $n>1$can be written in the form $p_1^{r_1},p_2^{r_2},p_3^{r_3},\dots ,p_t^{r_t}$, with the $p_j$distinct positive primes and every $r_j$.

Short answer

The required proof is done by Corollary 1.9.

Step-by-step solution

Statement of Corollary 1.9

According to Corollary 1.9, every integer $n$greater than 1 can be written in one and only one way in the form $n=p_1{p}_2\dots p_n$,where $p_i$s are positive primes such that $p_1\le p_2\le p_3\le \dots \le p_n$.

Proof

It is given that $n>1$, then by using corollary 1.9, for each positive primes $q_i$, such that the decomposition can be $n=q_1\dots q_s$.

By grouping all the similar primes, we can get the decomposition as

$n=p_1^{r_1}{p}_2^{r_2}{p}_3^{r_3}\dots p_t^{r_t}$.

Here,each $p_j$is distinct and equal to one of the $q_j$'s.

This implies $r_j>0$.

Hence, the given statement is proved.