Question

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:

Hint:

When minimizing a multi-digit number under digit constraints: \begin{itemize} \item First fix which digits must appear (here, two composites and one prime), \item Then choose the smallest possible digits, \item Finally arrange them in ascending order for the minimum value. \end{itemize}

Answer 1

Correct Answer: 6

Approach: First filter the legal digit pool, then build the smallest number greedily (smallest hundreds digit, then tens, then units), respecting the "exactly one prime digit" rule. Finally count factors.

Step 1 (legal digits): Digits are non-zero, so from \(\{1,\dots,9\}\). Remove perfect squares \(1, 4, 9\). Pool \(= \{2, 3, 5, 6, 7, 8\}\). Among these, primes are \(\{2, 3, 5, 7\}\) and non-primes are \(\{6, 8\}\).

Step 2 (the constraint bites): We need 3 distinct digits with exactly one prime. So two digits must be non-prime, but only \(6\) and \(8\) are non-prime in the pool. Hence the number must use both \(6\) and \(8\), plus exactly one prime.

Step 3 (minimise): The smallest prime available is \(2\). So the digit set is \(\{2, 6, 8\}\). Arrange ascending for the smallest 3-digit value: \(N = 268\). (Trying \(2\) in the hundreds place is best; \(268 < 286 < 628\dots\))

Step 4 (count factors): \(268 = 2^2 \times 67\). Number of factors \(= (2+1)(1+1) = 6\).

Answer: \(\boxed{6}\)

Answer 2

Approach: Sort the digits 1-9 into the relevant categories first (perfect squares, primes), see which digits are actually available, and build the smallest number from what's left.

Step 1: Identify forbidden and special digits. Perfect squares among 1-9: \(1, 4, 9\), all forbidden. Primes among the remaining digits \(\{2,3,5,6,7,8\}\): \(2,3,5,7\). Non-prime among them: only \(6\) and \(8\).

Step 2: Only one digit can be prime. Since exactly one digit is prime and the other two must be non-prime, non-square, distinct digits, and only \(6\) and \(8\) qualify as non-prime options, both must be used as the two non-prime digits. The third digit is whichever prime from \(\{2,3,5,7\}\) is chosen.

Step 3: Minimise \(N\). To keep \(N\) smallest, pick the smallest available prime, \(2\), and arrange all three digits \(\{2,6,8\}\) in increasing order: \(N = 268\).

Step 4: Factorise. \(268 = 2^2 \times 67\). Number of factors \(= (2+1)(1+1) = 6\).

\[ \boxed{6} \]