Question

Prove Corollary 1.9.

Short answer

Corollary 1.9 is proved.

Step-by-step solution

Statement of Corollary

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

Prove the corollary

$p_i$It is given that $n>1$, where it can be written as the product of its primes factors. $n=p_1{p}_2{p}_3\dots p_r$for any fixed integer $r$in an identical way to reordering for which each is well defined.

Since $n>1$, each $p_i$is positive.

According to the fundamental theorem of Arithmetic, there existaunique when we reorder all the factors.

So, there is a unique $p_i$such that$p_i\le p_{i+1}$ for $1\le i<r$.

So, the formed decomposition can be assumed to be positive, as $n$is positive in all primes, and there is a unique orderingsothat the product of all the primes will appear in increasing order, that is $p_1\le p_2\le p_3\le \dots \le p_n$.

Hence, Corollary 1.9 is proved.