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Question:

How many pairs (a, b) of positive integers are there such that a ≤ b and ab = 4²⁰¹⁷?

Explanation:

Given, a × b = 4²⁰¹⁷ = 2⁴⁰³⁴

Here, a and b and integers and will be some power of 2 since RHS is 2⁴⁰³⁴.

Let’s say a = 2^x and b = 2^y. [x and y are non-negative integers]

Also, a ≤ b, Hence, x ≤ y.

Now, 2^x × 2^y = 2⁴⁰³⁴

Since x ≤ y, x ≤ 4034/2 = 2017

The values x can take is 0, 1, 2, …. till 2017.

Hence, x can take a total of 2018 values.

Hence, option (d).