n is an integer.

(n-2) is divisible by 3 and 5,

(n-3) is divisible by 8,

What is the smallest possible value of n??

My solution is the following:

(n-2)=15k, where k is an integer, since it is divisible by 3 and 5

So (n-3)=15k-1 is divisible by 8, k=1, 2, 3, ...

The smallest number k such that (n-3) is divisible by 8 is k=7 (*)

i.e. n=107

Is there any other ways to solve this without listing all the possible values of (n-3) in step (*) and determine

whether (n-3) is divisible by 8.

Because if the smallest k is very big, then I will have a long list of numbers b4 I can get the answer.

Anyone can help?