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.