Answer to Question #350750 in Abstract Algebra for Dron

Question

Show that if a nd b are positive untegers, then ab=LCM(a,b)*GCD(a,b)

Answer 1

Let $GCD(a, b) = k$ and $LCM(a, b) = l$. Then $a=ka_1$ ad $b=kb_1$ where $GCD(a_1,b_1)=1$ and $ab=k^2a_1b_1$.

By defenition $a|l$ and $b|l$, moreover ig there exists an integer $s$ such that $a|s$ and $b|s$, then $l|s$.

Claim $l=ka_1b_1=ab_1=a_1b$. Indeed we have $a|ka_1b_1$ and $b|ka_1b_1$. Assume that there exists an integer $t$ such that $a|t$ and $b|t$. Then $t=ak_1$ and $t=bk_2$. We have $t=ak_1=bk_2=ka_1k_1=kb_1k_2$. It follows that $a_1k_1=b_1k_2$. Since $a_1$ and $b_1$ are relatively prime we have $a_1|k_2$ and $b_1|k_1$. The $k_2=a_1c$ and $k_1=b_1u$. Then we have $a_1k_1=a_1b_1u=b_1a_1c$ if follows that $u=c$ and $t=ak_1=ka_1b_1c$ hence $l=ka_1b_1|t$.