### How do you find all positive integers N such that the product 2029 x N has exactly four divisors?

2029 is a prime number. So let N be any prime number other than 2029. Then 2029*N has the four factors 1, 2029, N and 2029*N. [If N = 2029, then 2029 and N are the same and you have only 3 factors.] So in theory you can find N. However, since there are infinitely many prime numbers, you cannot find them all.

### How do you find all positive integers N such that the product 2029 x N has exactly three divisors?

There is only 1 value of N that satisfies 2029 x N has exactly three divisors: N = 2029 To have exactly three divisors, the number must be the square of a prime number. 2029 is a prime number with exactly 2 divisors (1 and 2029). Thus the only number with exactly three divisors of which two are 1 and 2029 is 20292 (= 4116841), making N = 2029.