HCF and LCM of two numbers are same. Find the difference between the two numbers.

Question

Q: HCF and LCM of two numbers are same. Find the difference between the two numbers.

solution: HCF and LCM of two numbers can be same only if two numbers are equal. Hence their difference will be zero.

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this was the answer in a book I saw. I did not understand it
What can be an alternative and mathematical proof for this?

Answer 1

Let $a$ and $b$ two positive integers. Then $\gcd(a,b)$ is a common divisor of $a$ and $b$ and therefore $\gcd(a,b)\leq \min(a,b)$. On the other hand $\text{lcm}(a,b)$ is a common multiple of $a$ and $b$ and therefore $\text{lcm}(a,b)\geq \max(a,b)$. Hence $$\gcd(a,b)\leq \min(a,b)\leq \max(a,b)\leq \text{lcm}(a,b)$$ (gcd is the same of hcf). So if $\gcd(a,b)=\text{lcm}(a,b)$ then $\min(a,b)=\max(a,b)$, that is $a=b$.

Answer 2

Let $\gcd (a, b) = \text{lcm} (a, b) = d$ $\gcd (a, b) = d \implies d\mid a\,\text{and}\, d\mid b$ $\text{lcm}(a, b) = d \implies a\mid d\,\text{and}\,b\mid d$ $d\mid a,\; a\mid d\implies a = d$ and $d\mid b,\; b\mid d \implies b = d.$ Hence $$a=b$$