Analysis:
Given that the sum of the number of factors of the number and N² is 34, we can represent this mathematically as:
Number of factors of N + N² = 34
Let's break this down step by step to find the distinct numbers that satisfy this condition.

Possible Values of N:
To find the possible values of N that satisfy the given condition, we need to consider the factors of 34. The factors of 34 are 1, 2, 17, and 34.

Calculating the Number of Factors:
For each of the factors of 34, we need to calculate the number of factors of N. This can be done by considering the prime factorization of N.
For example, if N = 1, the number of factors is 1.
If N = 2, the number of factors is 2.
If N = 17, the number of factors is 2.
If N = 34, the number of factors is 4.

Distinct Numbers:
From the above analysis, we can see that there are 3 distinct numbers that satisfy the given condition: 2, 17, and 34.
Therefore, there are 3 such distinct numbers N that satisfy the condition where the sum of the number of factors of the number and N² is 34.