JEE Maths Trick: Find Remainder of Huge Powers Mod 7 in Seconds! 🔥

 

❓ Question

Find the remainder when

$$((64{)}_{64}{)}^{64}$$

is divided by 7.

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✍️ Short Solution

This is a base + remainder theorem question.
JEE ka golden rule yaad rakho 👇
👉 Base-number ko pehle decimal me convert karo, phir modulo apply karo.

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🔹 Step 1 — Convert base-64 number to decimal

$$(64)_{64} = 6\times 64 + 4$$
$$= 384 + 4 = 388$$

So the expression becomes:

$$388^{64}$$

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🔹 Step 2 — Reduce base modulo 7

Instead of handling big powers, take modulo early:

$$388 \bmod 7 = 388 - 7\times 55 = 388 - 385 = 3$$

So,

$$388^{64} \equiv 3^{64} \pmod{7}$$

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🔹 Step 3 — Use Fermat’s Little Theorem

Since 7 is prime and $\gcd(3,7)=1$:

$$3^6 \equiv 1 \pmod{7}$$

Now reduce the exponent:

$$64 \bmod 6 = 4$$

So,

$$3^{64} \equiv 3^4 \pmod{7}$$

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🔹 Step 4 — Final calculation

$$3^4 = 81$$
$$81 \bmod 7 = 4$$

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✅ Final Answer

$$\boxed{4}$$

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