CAT 2025 Slot 1 · Quant · Question 5
CAT 2025 Slot 1 previous year question on Number of Factors, with the answer and a step-by-step solution.
Answer & solution
Answer: 6
First narrow the usable digits. They are non-zero, and 1, 4 and 9 are perfect squares, so the allowed digits are 2,3,5,6,7,8.
Among these the primes are 2,3,5,7 and the non-primes are just 6 and 8.
Exactly one digit may be prime, so the other two must be non-prime — and since only 6 and 8 are available, both must be used. The digits are 6, 8 and one prime.
To make N smallest, pick the smallest prime, 2, and arrange the digits in increasing order: N=268.
Finally 268={2}^{2}\times 67, so the number of factors is \left( 2+1 \right)\left( 1+1 \right)=6.
The step people miss is that 6 and 8 are forced; after that the rest is quick.
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