3-Digit Number Properties and Factors

Allowed non-zero, distinct digits that are NOT perfect squares ($1, 4, 9$ excluded): $\{2, 3, 5, 6, 7, 8\}$. Primes in this set: $\{2, 3, 5, 7\}$. Non-primes in this set: $\{6, 8\}$.

Since exactly one digit is prime and the other two are non-prime, the two non-prime digits MUST be $6$ and $8$. The single prime digit can be selected from $\{2, 3, 5, 7\}$.

To minimize $N$, we want the smallest hundreds digit. Choosing prime $2$ as the hundreds digit gives $N = 268$. Prime factorization of $268 = 2^2 \times 67$. Number of factors = $(2+1) \times (1+1) = 3 \times 2 = 6$.