CAT 2025 Slot 1 QA Question 7

"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 __."

Alright, so the question is about a 3-digit number N with certain conditions on its digits. We find the number of factors of the smallest such N.

First, let me break down the conditions:

So, to recap, the digits must be from {2, 3, 5, 6, 7, 8}, but with the following constraints:

Let me list all possible digits:

Digits allowed: 2, 3, 5, 6, 7, 8.

But primes among these are 2, 3, 5, 7. Non-primes are 6, 8.

So, the digits must include exactly one prime and two non-primes (6 or 8). Also, all digits must be distinct and non-zero.

Our goal is to find the smallest possible 3-digit number N satisfying these conditions and then find the number of factors of N.

So, first, let's find the smallest possible N.

To get the smallest 3-digit number, we want the hundreds digit to be as small as possible, then the tens digit, then the units digit.

But we have constraints on the digits:

Wait, so we have to choose one prime digit and two non-prime digits (6 or 8). So, the hundreds digit could be a prime or a non-prime, but we need to ensure that only one digit is prime.

So, let's think about how to distribute the prime digit.

Then, the other two digits must be non-prime (i.e., 6 or 8). So, the number would look like P 6 8 or P 8 6, where P is a prime digit.

But we need the smallest possible number, so P should be the smallest prime digit, which is 2.

So, the number would be 2 6 8 or 2 8 6. The smaller one is 268.

Then, one of the other digits (tens or units) must be prime. Since we need the smallest number, perhaps having the prime digit in the tens place would be better than in the units place because the tens place has a higher weight.

But .

If the hundreds digit is 6 or 8, which is non-prime, then we need one of the other digits to be prime.

So, for example, 6 _ _. The smallest prime is 2, so 6 2 _, but then the units digit has to be the other non-prime, which is 8. So the number would be 628.

Alternatively, 6 _ _, with the prime digit in the units place: 6 _ 2, but then the tens digit would have to be 8, making it 682, which is larger than 628.

Similarly, if hundreds digit is 8, the number would be larger, so 8 _ _ would be larger than 6 _ _.

So, comparing the two cases:

So, 268 is smaller.

But wait, is 268 the smallest possible?

Wait, let's check if there are other possibilities where the hundreds digit is prime, but the other digits are arranged to make a smaller number.

If hundreds digit is 2, the smallest prime, then the next digits should be as small as possible. The non-prime digits are 6 and 8. So, the smallest arrangement is 268.

Is there a way to have a smaller number with hundreds digit 2, but with a smaller digit in the tens place?

But the non-prime digits are only 6 and 8, so 268 is the smallest possible in this case.

Wait, but hold on. Are there any other non-prime digits besides 6 and 8? Let me check.

Digits allowed: 2,3,5,6,7,8.

Non-prime digits are 6 and 8, as 2,3,5,7 are primes, and 1,4,9 are excluded.

So yes, only 6 and 8 are non-prime digits.

Therefore, in Case 1, the smallest number is 268.

In Case 2, the smallest number is 628.

So, 268 is smaller.

But wait, is 268 the smallest possible? Let me think again.

Is there a way to have a hundreds digit as 2, and then have a smaller digit in the tens place? But the non-prime digits are 6 and 8, which are both larger than 2. So, 268 is the smallest.

Wait, but what about if the hundreds digit is 2, and then the tens digit is 6, and the units digit is 8? That's 268.

Alternatively, if the hundreds digit is 2, tens digit is 8, units digit is 6:

So, 268 is the smallest possible number.

Wait, but hold on. Let me check if 268 satisfies all conditions.

Digits: 2, 6, 8.

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