Question

divide the number 60 into 2 parts so that if one part is divided by 6 and the other part is divided by 8 the sum of the answer will be 9. both a and b are integers

Answer 1

If we divide  60 into say hmmm two integers, and those integers are say "a" and "b", then  a + b = 60, therefore, a = 60 - b.

$\bf \stackrel{\textit{one part is divided by 6}}{\cfrac{a}{6}}~~+~~\stackrel{\textit{the other part is divided by 8}}{\cfrac{b}{8}}~~=~~9 \\\\\\ \cfrac{60-b}{6}~~+~~\cfrac{b}{8}=9\impliedby \begin{array}{llll} \textit{let's multiply both sides by}\\ \textit{the LCD of 24} \end{array} \\\\\\ 24\left( \cfrac{60-b}{6}~~+~~\cfrac{b}{8} \right)=24(9)\implies 240-4b+3b=216 \\\\\\ -b=-24\implies b=\cfrac{-24}{-1}\implies b=24$

what's the first integer?  well, a = 60 - b.