Q93 / 180 · COMEDK 2023 (Memory Based) · Mathematics

$2^{3 n}-7 n-1$ is divisible by

  • A
    64
  • B
    36
  • Correct answer
    C
    49
  • D
    25

Solution

$\begin{aligned} & \text { Let } P(n)=2^{3 n}-7 n-1 \\ & \Rightarrow \quad P(1)=2^{3(1)}-7(1)-1=8-8=0 \\ & \Rightarrow \quad P(2)=2^{3(2)}-7(2)-1=64-15=49 \\ & \end{aligned}$ $P(1)$ and $P(2)$ are divisible by 49 . Let $P(k)=2^{3 k}-7 k-1=49 t$, where $t$ is an integer Now, $\begin{aligned} P(k+1) & =2^{3(k+1)}-7(k+1)-1=2^{3 k} \cdot 2^3-7 k-7-1 \\ & =8\left(2^{3 k}-7 k-1\right)+49 k \\ & =8(49 t)+49 k \\ & =49(8 t+k), \text { where } 8 t+k \text { is an integer } \end{aligned}$ Thus, $2^{3 n}-7 n-1$ is divisible by 49 .

Hence, the correct answer is option: 'C'.

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