Q28 / 75 · JEE Main 2025 (7 Apr Shift 1) · Mathematics

The remainder when $\left((64)^{(64)}\right)^{(64)}$ is divided by 7 is equal to

  • A
    4
  • Correct answer
    B
    1
  • C
    3
  • D
    6

Solution

Let $\mathrm{N}=\left((64)^{64}\right)^{64}$ $\mathbf{N}=(64)^{64^2}$ $\mathbf{N}=(1+63)^{64^2}, \text { let } 64^2=\mathrm{n}$ Expanding by binomial $\begin{aligned} & \mathrm{N}=(1+63)^{\mathrm{n}}=1+{ }^{\mathrm{n}} \mathrm{C}_1 63+{ }^{\mathrm{n}} \mathrm{C}_2(63)^2+\ldots . \\ & =1+63 \lambda=1+7(9 \lambda) \end{aligned}$ Remainder when divided by 7 is 1

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

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