Question

The GCD(a, b) = 9, the LCM(a, b)=378. Find the least possible value of a+b.

Answer 1

$\mathrm{gcd}(a,b)=9\implies9\mid a\text{ and }9\mid b\implies9\mid a+b$

which means there is some integer $k$ for which $a+b=9k$.


Because $9\mid a$ and $9\mid b$, there are integers $n_1,n_2$ such that $a=9n_1$ and $b=9n_2$, and


$\mathrm{lcm}(a,b)=\mathrm{lcm}(9n_1,9n_2)=9\mathrm{lcm}(n_1,n_2)=378\implies\mathrm{lcm}(n_1,n_2)=42$

We have $42=2\cdot3\cdot7$, which means there are four possible choices of $n_1,n_2$:

1, 42
2, 21
3, 14
6, 7

which is to say there are also four corresponding choices for $a,b$:

9, 378
18, 189
27, 126
54, 63

whose sums are:

387
207
153
117

So the least possible value of $a+b$ is 117.