Question

examine whether (15)n can end with digit 0 for any n € N

Answer 1

If the number (15)^n, for any n, belonging to natural numbers N, were to end with the digit 0, then it would be divisible by 10.

That is, the prime factorization of 15^n would contain the primes 5 & 2 positively.

But this is not possible, because 15^n = (3 x 5)^n

So the only primes in the factorizayion of 15^n , are 3 & 5. We have 5 but there is no 2.

So, the uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no other primes in the factorization of 15^n

So, there is no natural number n for which 15^n ends with the digit zero.

Answer 2

As we know that

15n = (3 x 5)n

  =  3n x 5n

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, 15n will never end in zero as 15 doesn't has 2 as a factor.

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