Make 10!

We are looking for positive integers $x$ such that $82 = nx + 10$ for some integer $n$ , or in modular form, $82 \equiv 10 \pmod{x}$ . We are thus looking for $x \gt 10$ such that $72 \equiv 0 \pmod{x}$ . Such $x$ are the positive divisors of $72$ that exceed $10$ , which are $12, 18, 24, 36$ and $72$ , a total of $\boxed{5}$ numbers.

> Since we have remainder of 10 so the number we will be dealing upon is $82 - 10 => 72$ now we have to make factors of 72 . by prime factorization method we get 12 total factors . Since we only want the number greater than 10 to have a remainder of 10 so we will remove $1,2,3,4,6,8,9$ , after we had subtracted 7 ( because we only want factors greater than 10 ) our answer implies $12-7 ==>$ 5