Question

How many pairs \((a, b)\) of positive integers are there such that \(a≤b\) and \(ab=4^{2017}\) ?

• A. 2017
• B. 2019
• C. 2020
• D. 2018

Answer 1

The Correct Option is D

To find the number of pairs \((a, b)\) of positive integers such that \(a \leq b\) and \(ab = 4^{2017}\), we start by examining the factorization of \(4^{2017}\):

\[4^{2017} = (2^2)^{2017} = 2^{4034}\]

For \(a\) and \(b\) to be pairs of positive integers satisfying \(ab = 2^{4034}\), let \(a = 2^x\) and \(b = 2^y\) with \(x \leq y\). Then, we have:

\[a \times b = 2^x \times 2^y = 2^{4034}\]

This means:

\[x + y = 4034\]

Since \(a \leq b\), we have:

\[2^x \leq 2^y \implies x \leq y\]

Substituting \(x + y = 4034\) into \(x \leq y\), we get:

\[x \leq \frac{4034}{2} = 2017\]

Therefore, \(x\) ranges from 0 to 2017. Once \(x\) is selected, \(y\) is uniquely determined by \(y = 4034 - x\), and it will automatically satisfy \(y \geq x\) due to how \(x\) is constrained. Thus:

The number of possible values for \(x\) is:

\[2017 - 0 + 1 = 2018\]

Therefore, there are 2018 pairs \((a, b)\) such that \(a \leq b\) and \(ab = 4^{2017}\).

Correct Answer: 2018

Answer 2

Given, 

\( ab = 4^{2017} = (2^2)^{2017} = 2^{4034} \)

Step 1: Total number of positive integer factors of \( ab = 2^{4034} \)

Since it has only one prime factor, total number of factors = \( 4034 + 1 = 4035 \)

Step 2: We want to find the number of ordered or unordered pairs \( (a, b) \) such that:

• \( a \times b = 2^{4034} \)
• \( a \leq b \) (i.e., count unique pairs without repetition)

Every factor pair \( (a, b) \) corresponds to a product \( ab = 2^{4034} \), and each factor divides the number.

Total such factor pairs = \( \left\lfloor \frac{4035}{2} \right\rfloor = 2017 \)

But since \( ab = 2^{4034} \) is a perfect square, one of the factor pairs will be \( ( \sqrt{2^{4034}}, \sqrt{2^{4034}} ) \)

\( \sqrt{2^{4034}} = 2^{2017} \) (This is the repeated middle factor)

So, we count this middle pair only once and add it to the 2017 distinct pairs.

Total pairs (a, b) with \( a \leq b \) and \( ab = 2^{4034} \) = 2017 + 1 = 2018

∴ Correct Answer: Option (D) — 2018