CAT 2023 Set-3 | Quantitative Aptitude | Question: 2

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 $n+m$ is

• A. $44$
• B. $16$
• C. $42$
• D. $14$

   

Answer 1

Between 8^m  and 8ⁿ   there should be exactly 41 integers which can be expressed as powers of 2.

8^m  = 2^3m  and 8ⁿ = 2³ⁿ 

Since we want the smallest possible value of m + n, so m and n values should be as minimum as possible.

let's take m = 1.

 2^3m  =  2³, now after  2³   from   2⁴  to   2⁴⁴ so exactly a total of  41 integers came and after  2⁴⁴,  2⁴⁵ came.

so,  2³ⁿ = 2⁴⁵

which makes 3n = 45 -> n = 15

so m + n = 1 + 15 = 16.

Option B is the correct answer.