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JEE Main 20257 AprilMorning Shift
Exam
JEE Main
Year
2025
Date
7 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2025 · 7 April · Morning Shift · Mathematics · Binomial Theorem

The remainder when ((64)^ (64) )^ (64) is divided by 7 is equal to

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

Options

  1. A4
  2. B1
  3. C3
  4. D6

Correct answer

B. 1

Step-by-step solution

Let N=((64)64)64N=\mathrm{N}=\left((64)^{64}\right)^{64} \mathbf{N}=(64)^{64^2} N=\mathbf{N}=(1+63)^{64^2},  let 642=nExpandingbybinomial\text{ let } 64^2=\mathrm{n} Expanding by binomialN=(1+63)n=1+nC163+nC2(63)2+….=1+63λ=1+7(9λ)\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}Remainderwhendividedby7is1Remainder when divided by 7 is 1
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