In a 3-digit number $N$, the digits are non-zero and distinct such that none of the digits is a perfect square, and only one of the digits is a prime number. Then, the number of factors of the minimum possible value of $N$ is:
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Available digits (non-zero, non-squares): $\{2, 3, 5, 6, 7, 8\}$.
Prime candidates: $\{2, 3, 5, 7\}$. Non-primes: $\{6, 8\}$.
Smallest $N$ needs two digits from $\{6, 8\}$ and one from $\{2, 3, 5, 7\}$.
Smallest combination: $2, 6, 8 \implies N = 268$.
Step 2: Apply Prime constraints
Primes in set: $\{2, 3, 5, 7\}$.
Non-primes in set: $\{6, 8\}$.
The number $N$ must have 1 prime digit and 2 non-prime digits. To minimize $N$, we must use both non-prime digits ($6$ and $8$) and the smallest prime digit ($2$).
Step 3: Find Minimum $N$ and its factors
Digits are $\{2, 6, 8\}$. Smallest 3-digit number formed is $N = 268$.
Prime factorization of $268$:
$268 = 2 \times 134 = 2^2 \times 67$.
Number of factors $= (2+1) \times (1+1) = 3 \times 2 = \mathbf{6}$.
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