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Binomial Theorem question

2025 · 7 Apr · Shift 1 · Q34

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Binomial Theorem
2025 · 7 Apr · Shift 1 · Q34

Binomial Theorem question

2025 · 7 Apr · Shift 1 · Q34

JEE MainMathematicsBinomial TheoremMCQ+4 / −1

The remainder when

\left((64)^{(64)}\right)^{(64)}

is divided by 7 is equal to

A
4
B
6
C
3
D
1

View written solutionFree

Correct answer: D

Interpret the expression

We need the remainder when $\left((64)^{64}\right)^{64}$ is divided by $7$ .

Using the law of exponents, $\left((64)^{64}\right)^{64} = 64^{64\cdot 64} = 64^{4096}.$

So we must find: $64^{4096} \pmod{7}.$

Reduce the base modulo 7

Since $64 \equiv 1 \pmod{7}$ because $64 = 7\cdot 9 + 1$ .

Therefore, $64^{4096} \equiv 1^{4096} \pmod{7}.$

Compute the result

Now, $1^{4096} = 1.$

Hence, $64^{4096} \equiv 1 \pmod{7}.$

So the remainder is: $\boxed{1}$

Check options

A: $4$ ❌
B: $6$ ❌
C: $3$ ❌
D: $1$ ✅

Therefore, the correct option is D.

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