Least Common Multiple and Greatest Common Divisor

Question

Prove that if $\mathop{\mathrm{lcm}}( a, b) + \gcd(a, b) = a+b$, $a$ divides $b$ or $b$ divides $a$.

This problem seemed simple at first, however I cannot figure out a way to prove this. If I assume both parts of the statement, I can show it is true, but that is not a valid proof method.

Answer 1

Let $d = \gcd(a, b)$, and let $a = dA$ and $b = dB$.

Then, since $\operatorname{lcm}(a, b) =\frac{ab}{\gcd(a, b)} =\frac{ABd^2}{d} =ABd $, $d+ABd =Ad+Bd $, so $1+AB =A+B $ or $0 =AB-A-B+1 =(A-1)(B-1) $.

Therefore either $A=1$ or $B=1$. If $A=1$, then $a = dA = d$ divides $b = dB$. Similarly, if $B = 1$, $b$ divides $a$.