Number System Question 7 - CAT 2020

Step-by-Step Explanation

Let's convert the given equation to base 2 to find its total factors:

ab = 4²⁰¹⁷ = ( 2² )²⁰¹⁷ = 2⁴⁰³⁴

The total number of factors for a prime power pᵏ is given by the formula k + 1:

Total Factors = 4034 + 1 = 4035 factors

We want to find pairs (a,b) such that a × b = 2⁴⁰³⁴ and a ≤ b.

Every factor can be paired with another factor to multiply to the total value. Since the total number of factors (4035) is an odd number, the value 2⁴⁰³⁴ is a perfect square. This means there is exactly 1 pair where the factors are equal:

a = b = 2²⁰¹⁷

The remaining 4035 - 1 = 4034 factors can be grouped into distinct pairs where a < b:

Distinct Pairs = (4034)/(2) = 2017 pairs

Total pairs where a ≤ b = 2017 + 1 = 2018

Correct Option: A

💡 FundaMakers Tip:

For any product equation ab = N where a ≤ b, the total number of unique factor pairs is always given by (F + 1)/(2) (if the number of factors F is odd). Converting to a prime base immediately gives the factor count and the final answer.