## Is gcd(a, b) * C(b,a)/b always an integer?

$gcd \, (\, 2, \, 3 \, ) \; \times \; \dbinom{\, 3 \,}{\, 2 \,}\, /\, 3 \; = \; 1$

$gcd \, ( \, 2, \, 8 \, ) \; \times \; \dbinom{\, 8\, }{\, 2\, }\, /\, 8 \; = \; 7$

$gcd \, ( \, 2, \, 9 \, ) \; \times \; \dbinom{\, 9\, }{\, 2\, }\, /\, 9 \; = \; 4$

$gcd \, ( \, 4, \, 6 \, ) \; \times \; \dbinom{\, 6\, }{\, 4\, }\, /\, 6 \; = \; 5$

Is   $gcd \,(\, a, \, b \,) \; \times \; \dbinom{\, b\, }{\, a\, }\, /\, b$   always an integer for every pair $(a, b)$   of postive integers
with   $1 \leq a \leq b$