[Math] Prove that if $a$ and $b$ are positive integers satisfying $\gcd(a,b)=\operatorname{lcm}(a,b)$,then $a=b$

Question

Prove that if $a$ and $b$ are positive integers satisfying $\gcd(a,b)=\operatorname{lcm}(a,b)$,then $a=b$.

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Since the formula for two positive integers $a,b$ is $\operatorname{lcm}(a,b)=\frac{ab}{\gcd(a,b)}$

As $\gcd(a,b)=\operatorname{lcm}(a,b)$, so $\gcd(a,b)\gcd(a,b)=ab$

I am stuck here.I dont know how to prove further.Please help.

Answer 1

I would probably do it differently, but we can use your equation as a start.

Note that if $d=\gcd(a,b)$, then $d\le b$. So $ab=d^2\le b^2$. It follows that $a\le b$. A similar argument shows that $b\le a$. It follows that $a=b$.