Maths Q7 – JEE-MAIN 2025

The remainder when

(64)^{(64)^{(64)}}

is divided by 7 is equal to

Clue 1 of 3: Simplify the base

First, find the remainder of the base when divided by the modulus.

🥷

Ninja Strategy•Base Simplification

First, simplify the base $64$ modulo $7$ . Since $64 \equiv 1 (\mod 7)$ , any positive integer power of $64$ will also be $1 (\mod 7)$ .

🧠 Full Solution Path

Step 1: Simplify the base modulo 7✦ Active

First, find the remainder of the base $64$ when divided by $7$ .

64 = 9 \times 7 + 1

So, $64 \equiv 1 (\mod 7)$ .

Step 2: Apply the modular exponentiation property○ Expand

Since the base is congruent to $1 (\mod 7)$ , any positive integer power of this base will also be congruent to $1 (\mod 7)$ . The exponent $(64)^{(64)}$ is a positive integer.

(64)^{(64)^{(64)}} \equiv 1^{(64)^{(64)}} (\mod 7)

Therefore, $1^{(64)^{(64)}} \equiv 1 (\mod 7)$ .

💡 Teacher's Secret Hint

Remember that $1^{k} = 1$ for any positive integer $k$ .

Step 3: State the final remainder○ Expand

The remainder when $(64)^{(64)^{(64)}}$ is divided by $7$ is $1$ .

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Maths Question 7 – JEE-MAIN 2025

Clue 1 of 3: Simplify the base

🧠 Full Solution Path