CAT 2023 Question Paper | Quant Slot 3

CAT 2023 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 6 : The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is

Video Explanation

Explanatory Answer

If an integer N can be prime factorized as N = \( p ^ { a } q ^ { b } \), then the number of factors of N is \( ( a + 1 ) \times ( b + 1 ) \).
Therefore, for a number N to have 15 factors, its prime factorization should be of the form \( N = p ^ { 14 } \& N = p ^ { 2 } q ^ { 4 } \).
The smallest number of the form \( N = p ^ { 14 } = 2 ^ { 14 } = 16,384 \)
The smallest number of the form \( N = p ^ { 2 } q ^ { 4 } = 3 ^ { 2 } 2 ^ { 4 } = 144 \)
The second smallest number of the form \( N = p ^ { 2 } q ^ { 4 } = 2 ^ { 2 } 3 ^ { 4 } = 324 \)
Therefore, the sum of the two smallest numbers is 144 + 324 = 468

The question is " The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is "

Hence, the answer is '468'