As a is the generator so order of a =n i.e o(a)=10 in our case.
We know a^10=e. we know that a^8 is not an identity.Let us say a^8=b.
Now we need to find order of b.
o(b) must divide order of group.It can be 1,2,5(+ve divisors of 10). Order 1 is not possible as b is not identity element.
2,5,10 are left options.
let o(b) as x.We need value of x in b^x=e.
put b as a^8.
Now we need (a^8)^x=e
.Try with x=2.it becomes a^16.
a^16= a^10*a^6. ,which means e*a^6.Hence this cannot give identity
Try x=5
(a^8)^x=e.
means a^40 which is e. Hence 5 is our answer.
so this mean b^5=e and 5 is answer .As 5 is satisfying so no need to check for 10 as we need smallest +ve integer.
Please let me know if anything is wrong in the solution