Find all natural numbers N such that the remainder of 2017 divided by N is 17.

Find all natural numbers N such that the remainder of 2017 divided by N is 17. In your answer, indicate the number of such N.

Since when dividing 2017 by each number N, a remainder of 17 should be obtained, then 2017 can be represented as:

2017 = a * N + 17, where a is some natural number.

Subtract from both sides of formula 17:

a * N = 2000.

Since a and N are natural numbers, then all numbers N are divisors of 2000, and since the remainder of 2017 divided by N must be 17, these divisors must be greater than 17.

Factor 2000:

2 * 10 * 10 * 10 = 2 * 2 * 2 * 2 * 5 * 5 * 5.

Let’s compose all possible products from no more than four twos and three fives:

2 * 2 * 2 * 2 * 5 = 80;

2 * 2 * 2 * 2 * 5 * 5 = 400;

2 * 2 * 2 * 2 * 5 * 5 * 5 = 2000;

2 * 2 * 2 * 5 = 40;

2 * 2 * 2 * 5 * 5 = 200;

2 * 2 * 2 * 5 * 5 * 5 = 1000;

2 * 2 * 5 = 20;

2 * 2 * 5 * 5 = 100;

2 * 2 * 5 * 5 * 5 = 500;

2 * 5 * 5 = 50;

2 * 5 * 5 * 5 = 250;

5 * 5 = 25;

5 * 5 * 5 = 125.

Answer: 13.