As we know that

15^{n} = (3 x 5)^{n}

= 3^{n} x 5^{n}

The prime factorization of 15 doesn't have a 2 and 5 as its factor. So, its factorization will never end in 10 as a number ending in 10 must have a factors as 5 and 2. So, 15^{n} will never end in zero as 15 doesn't has 2 as a factor.