Question

N is a 3-digit number with non-zero digits. No digit is a perfect square and only 1 of the digits is a prime number. What is the number of factors of the smallest such number possible?

Hint:

To form the smallest number from a given set of digits, arrange them in ascending order. To find the largest number, arrange them in descending order. Always tackle number theory problems by breaking down the constraints one by one.

Answer 1

Correct Answer: 6

Step 1: Understanding the Question: 
The problem asks us to first identify the smallest 3-digit number 'N' that meets a specific set of criteria for its digits. Once we find this number, we need to calculate how many factors (divisors) it has. 
Step 2: Key Formula or Approach: 
To find the number of factors of an integer N, we first find its prime factorization, say \(N = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}\). The total number of factors is then given by the product of one more than each exponent: 
\[ \text{Number of Factors} = (a_1 + 1)(a_2 + 1)\cdots(a_k + 1) \] 
Step 3: Detailed Explanation: 
We need to find the smallest number N by carefully applying all the given conditions. 
Condition 1: Digits are non-zero. 
The possible digits are \(\{1, 2, 3, 4, 5, 6, 7, 8, 9\}\). 
Condition 2: No digit is a perfect square. 
The single-digit perfect squares are 1, 4, and 9. We must exclude these. 
The remaining allowed digits are\( \{2, 3, 5, 6, 7, 8\}\). 
Condition 3: Only 1 of the three digits is a prime number. 
From the allowed set \(\{2, 3, 5, 6, 7, 8\}\), let's identify the primes and non-primes. 
- Prime digits: \(\{2, 3, 5, 7\}\). 
- Non-prime (composite) digits: \(\{6, 8\}.\) 
The condition means our 3-digit number must be formed using exactly one digit from \(\{2, 3, 5, 7\}\) and two digits from \(\{6, 8\}. \)
Finding the smallest possible number N: 
To construct the smallest possible 3-digit number, we must: 
1. Choose the smallest possible digits that fit the criteria. 
2. Arrange these digits in ascending order. 
To get the smallest digits, we should choose the smallest prime (which is 2) and the two non-primes (which are 6 and 8). 
The set of digits we must use is \(\{2, 6, 8\}. \)
To form the smallest number with these digits, we place the smallest digit in the hundreds place, the next smallest in the tens place, and the largest in the units place. 
Smallest number N = 268. 
Finding the number of factors of N = 268: 
First, we find the prime factorization of 268. 
\[ 268 = 2 \times 134 \] \[ 268 = 2 \times 2 \times 67 \] \[ 268 = 2^2 \times 67^1 \] Here, the exponents are \(a_1 = 2\) and \(a_2 = 1\). 
Using the formula for the number of factors: 
\[ \text{Number of Factors} = (2 + 1)(1 + 1) = 3 \times 2 = 6 \]
 Step 4: Final Answer: 
The number of factors of the smallest such number (268) is 6.