Number System Question 1 - CAT 2025
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:
Step-by-Step Explanation
- Identify the eligible digits:
The digits must be non-zero and distinct. None of them can be perfect squares (1,4,9). The remaining allowable digits are:
2,3,5,6,7,8
Among these choices, the prime digits are 2,3,5,7 and the non-prime digits are 6,8.
- Minimize the value of N:
The number must contain exactly one prime digit and two non-prime digits. Since only two non-prime digits exist (6,8), both must be included in the number. To minimize N, select the smallest available prime digit, which is 2. Arranging the distinct digits 2,6,8 in ascending order yields:
Nₘᵢₙ = 268
- Find the number of factors:
Prime factorize 268 to find its total divisors:
268 = 4 × 67 = 2² × 67¹
Number of factors = (2 + 1) × (1 + 1) = 3 × 2 = 6
Correct Answer: 6
💡 FundaMakers Tip:
When minimizing a number with strict digital properties, always isolate the highly restricted sets first. Because the problem demands two non-prime numbers and we only have two options (6 and 8), they act as fixed anchors. The remaining value is determined entirely by picking the smallest available prime to occupy the highest place value.