CAT 2023 Slot 3 QA Question 7

Type-in-the-answer (no negative marking) · Number Systems · Factorisation · Try it, then check the answer and solution below.

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

TITA Answer:

⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: 468

Answer and solution

Answer: 468

📌 Core Concept & Formula

The number of factors of a natural number

N

with prime factorization

N = p_1^{a} \times p_2^{b} \times p_3^{c} \times \ldots

is given by:

\text{Number of factors} = (a+1)(b+1)(c+1)\ldots

For a number to have exactly 15 factors,

(a+1)(b+1)(c+1)\ldots = 15

🔢 Step-by-Step Solution

Factorize 15:

15 = 15 \times 1

15 = 5 \times 3

Determine possible forms of

N

Case 1:

N = p^{14}

where

p

is prime.

Case 2:

N = p^4 \times q^2

where

p

and

q

are distinct primes.

Find the smallest numbers for each case:

Case 1: Smallest

N = 2^{14} = 16384

Case 2: Smallest

N = 2^4 \times 3^2 = 16 \times 9 = 144

Identify the next smallest number in Case 2:

N = 3^4 \times 2^2 = 81 \times 4 = 324

Sum the two smallest numbers:

\[ 144 + 324 = 468

Keep going