Some Sum and Multiple

Let the two integers be $A$ and $B$ , with greatest common factor $g$ . Then for some coprime numbers $a$ and $b$ , $A = ga$ & $B = gb$ Hence, $A + B = g(a + b) = 216$ $(Eq. 1)$

But the product of the greatest common factor and least common multiple is the product of the numbers themselves, i.e., $(GCF)(LCM) = AB.$ Then,

$g(480) = (ga)(gb)$ or $480 = gab$ . $(Eq. 2)$

But from $Eq 1$ and $Eq 2$ , $a + b$ and $ab$ have no common factors . It then follows that

$g = GCF$ of $216$ and $480$

i.e., $g = GCF$ of $(2^3)(3^3) and (2^5)(3)(5)$ or $g = (2^3)(3) = 24.$

From here we have $a + b = \frac{216}{24}$ = 9 \and $ab = \frac{480}{24}$ = 20. Solving for $a$ and $b$ gives $4$ and $5$ , respectively. Hence, $A = 24(5) = 120$ and $b = 24(4) = 96$