What is the probability that the product of the integers from n-2 to n+2 are also a product of 10 if n∈{13,14,15,16,17}?

Question

What is the probability that the product of the integers from n-2 to n+2 are also a product of 10 if n∈{13,14,15,16,17}?

For Example
If n was 13 then the product would be 11*12*13*14*15


I want an answer that does not use brute force.

Answer 1

The probability is $1$, because every $5$ consecutive numbers must contain a multiple of $5$ and a multiple of $2$, whence the product is a multiple of $10$ (because $5$ and $2$ are co-prime).