Just 2014

If $2014$ leaves a remainder of $14$ when divided by $n$ , it means that $2014-14 = 2000$ leaves no remainder when divided by $n$ . In toher words, $n$ divides $2000$ . $2000 = 2^4 \times 5^3$ so there are $(4+1)(3+1) = 5(4) = 20$ factors of $2000$ . But note that $n>14$ . We see that there are $6$ factors of $2000$ less than $14$ namely - $1, 2, 4, 5, 8, 10$ . So $n$ can take on $20-6 = \boxed{14}$ values.

Did you mean "Find the number of positive integers $n$ such that when $2014$ is divided by $n$ , the remainder is $14$ ."?

If so, the answer is 14.

We see that $n > 14$ , otherwise the remainder will be less than $14$ .

And, by the remainder theorem, $2014 = nq + 14 \implies nq = 2000$ , where $q$ is an integer. Therefore, $n$ is a factor $2000$ .

Using these two facts, we find the total number of divisors of 2000 more than 14 is $\boxed{14}$ .