CAT 2020 Slot 3 QA Question 7: How many pairs $(a, b)$ of positive integers are there such that $a \leq b$ and $ab = 4^{2017}$?

Question

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

Accepted Answer

Prime factorize the product.
$4^{2017} = (2^2)^{2017} = 2^{4034}$
Every factor is of the form $2^r$, where $r$ can range from 0 to 4034.
If $a = 2^r$, then $b = 2^{4034-r}$.
The condition $a \leq b$ gives $r \leq 4034-r$, so $r \leq 2017$.
Therefore, $r$ can take values from 0 to 2017, giving 2018 p

How many pairs (a,b)(a, b)(a,b) of positive integers are there such that a≤ba \leq ba≤b and ab=42017ab = 4^{2017}ab=42017?

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