Question

Prove that $\left(a,b\right)=1$if and only if there is no prime $p$such that$p|a$ and $p|b$.

Short answer

It is proved that$\left(a,b\right)=1$ if and only if there is no prime $p$such that $p|a$and $p|b$.

Step-by-step solution

Necessary condition

Assume there are two integers, $a$and $b$;theirgcd will be 1 if there is no prime such that $p|a$and $p|b$.

Now, take $\left(a,b\right)=1$.Then there exists any prime such that the common divisor of $a$and $b$is $p$, where , which contradicts the greatest common divisor is 1. Hence, there is no prime$p$ with $p|a$and $p|b$.

Sufficient condition

Now assume that there is no prime $p$such that $p|a$and $p|b$. Then for any two integers $a$and $b$, $\left(a,b\right)\ne 1$. Let $\left(a,b\right)=d$.

Now, using the fundamental theorem of arithmetic, $d$is the product of primes. Then $p$will be a factor of $d$. Since $d|a$and$d|b$,then $p|a$and $p|b$that are contradictionsto the assumption, so the value of $d$should be 1.

Hence, it is proved that $\left(a,b\right)=1$if and only if there is no prime $p$such that $p|a$and $p|b$.