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

Question

Arun selected an integer \( x \) between 2 and 40, both inclusive. He noticed that the greatest common divisor of the selected integer \( x \) and any other integer between 2 and 40, both inclusive, is 1. How many different choices for such an \( x \) are possible?

Answer 1

Determining Numbers with 15 Factors

Given that a number \( N \) has 15 factors.

The factor count of 15 can be expressed as \( 15 = 1 \times 15 = 3 \times 5 \).

Therefore, the possible exponent combinations for the prime factorization of \( N \) are:

\( (p+1)(q+1) = 3 \times 5 \Rightarrow p = 2, q = 4 \)
This implies \( N \) can be represented as \( N = a^2 \cdot b^4 \) or \( N = a^4 \cdot b^2 \), where \( a \) and \( b \) are distinct prime numbers.

Scenario 1: \( N = 2^4 \times 3^2 \)

\( 2^4 = 16 \), \( 3^2 = 9 \)
\( \Rightarrow N = 16 \times 9 = 144 \)

Scenario 2: \( N = 2^2 \times 3^4 \)

\( 2^2 = 4 \), \( 3^4 = 81 \)
\( \Rightarrow N = 4 \times 81 = 324 \)

Sum of the Two Smallest Numbers Found:

\( 144 + 324 = 468 \)

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

The Correct Option is C

Solution and Explanation

Determining Numbers with 15 Factors

Scenario 1: \( N = 2^4 \times 3^2 \)

Scenario 2: \( N = 2^2 \times 3^4 \)

Sum of the Two Smallest Numbers Found:

Correct Answer: (C): \( \boxed{468} \)

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