The number of seashells collected by 13 boys and n girls is n2+10n-18. if each child collects exactly the same number of seashells, determine the value of n.

Say each child collects x shells. so, total shells collected are (n+13)x =n^2+10n-18

x=(n^2+10n-18)/(n+13)=[n(n+13)-3(n+6)]/(n+13)=n-[3(n+6)/(n+13)] = a natural number

for different values of n, we get different values of x.but we want n such that x is a natural number. we can observe by putting n=1,2,3... that n=8 gives x=8-2=6.

so, n=8 is certainly an answer.