[tex]\mathrm{gcd}(a,b)=9\implies9\mid a\text{ and }9\mid b\implies9\mid a+b[/tex]
which means there is some integer [tex]k[/tex] for which [tex]a+b=9k[/tex].
Because [tex]9\mid a[/tex] and [tex]9\mid b[/tex], there are integers [tex]n_1,n_2[/tex] such that [tex]a=9n_1[/tex] and [tex]b=9n_2[/tex], and
[tex]\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[/tex]
We have [tex]42=2\cdot3\cdot7[/tex], which means there are four possible choices of [tex]n_1,n_2[/tex]:
1, 42
2, 21
3, 14
6, 7
which is to say there are also four corresponding choices for [tex]a,b[/tex]:
9, 378
18, 189
27, 126
54, 63
whose sums are:
387
207
153
117
So the least possible value of [tex]a+b[/tex] is 117.
