CAT 2023 Slot 3 QA Question 4: Let n and m be two positive integers such that there are exactly 41 integers greater than $8^m$ and less than $8^n$, which can be expressed as powers of 2. Then, the smallest possible value of $n + m$

Question

Let n and m be two positive integers such that there are exactly 41 integers greater than $8^m$ and less than $8^n$, which can be expressed as powers of 2. Then, the smallest possible value of $n + m$

Accepted Answer

Write the powers of 8 as powers of 2.

$8^m = 2^{3m}$ and $8^n = 2^{3n}$

The powers of 2 strictly between these are
$2^{3m + 1}, 2^{3m + 2}, \ldots, 2^{3n - 1}$

So the number of such powers is
$(3n - 1) - (3m + 1) + 1 = 3n - 3m - 1$
$3n - 3m - 1 = 41$
$3(n - m) = 42$
$n - m = 14$

To minimize $n +

Let n and m be two positive integers such that there are exactly 41 integers greater than 8m8^m8m and less than 8n8^n8n, which can be expressed as powers of 2. Then, the smallest possible value of n+mn + mn+m is

Let n and m be two positive integers such that there are exactly 41 integers greater than $8^m$ and less than $8^n$ , which can be expressed as powers of 2. Then, the smallest possible value of $n + m$ is