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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Step-by-Step Solution
Pattern: this is a digit-property filtering question combined with minimize-then-compute — recognisable because the digits must satisfy several yes/no filters (non-zero, distinct, not a perfect square, exactly one prime), and then we must build the smallest number and analyse a property of it (its factor count).
Step 1 — List the allowed digit pool.
Digits are from to (non-zero). Perfect squares among these are — these are banned since "none of the digits is a perfect square."
Remaining allowed digits: .
Step 2 — Split this pool by primality.
Prime digits in the pool: .
Non-prime digits in the pool: (only two digits!).
Step 3 — Use the "only one prime digit" condition.
has 3 distinct digits, and exactly one of them is prime — so exactly two digits must be non-prime. But the non-prime pool has only two members: and . So both non-prime digits are forced: they must be exactly .
Step 4 — Choose the prime digit to minimize N.
The third digit is any one prime from . To make as small as possible, pick the smallest available prime digit: .
So the digit set is .
Step 5 — Arrange the digits to minimize the number.
With all three digits non-zero, the smallest 3-digit number from a given digit set is formed by placing digits in ascending order: hundreds < tens < units.
Step 6 — Verify all conditions.
Digits : non-zero ✓, distinct ✓, none is a perfect square (1,4,9 excluded) ✓, exactly one is prime (2 is prime; 6,8 are not) ✓.
Step 7 — Find the number of factors of 268.
(67 is prime).
Number of factors .
Trap: a student who doesn't notice that the non-prime pool has only two elements might try to freely choose 2 non-prime digits from a larger set, missing that is forced — or might forget that a perfect-square digit (like 1, which looks "small" and tempting for minimizing) is actually banned.