When $10^{100}$
is divided by 7 , the remainder is

Question

When $10^{100}$
 is divided by 7 , the remainder is

Accepted Answer

The problem requires finding the remainder when $$10^{100}$$ is divided by 7. Here's the step-by-step solution:

**Simplify Using Modular Arithmetic:**

- First, calculate $$10 \mod 7$$:
     $$
     10 \div 7 = 1 \, 	ext{remainder} \, 3
     $$
     Thus:
     $$
     10 \equiv 3 \pmod{7}
     $$

- Substitute $$10^{100}$$ with $$3^{100}$$ under modulo 7:
     $$
     10^{100} \mod 7 = 3^{100} \mod 7
     $$

**Cycle of Powers Modulo 7:**

- Compute successive powers of $$3$$ modulo $$7$$ to identify patterns:
     - $$3^1 \mod 7 = 3$$
     - $$3^2 \mod 7 = 9 \mod 7 = 2$$
     - $$3^3 \mod 7 = 6$$
     - $$3^4 \mod 7 = 18 \mod 7 = 4$$
     - $$3^5 \mod 7 = 12 \mod 7 = 5$$
     - $$3^6 \mod 7 = 15 \mod 7 = 1$$

- The powers repeat every six steps: $$3, 2, 6, 4, 5, 1$$.

**Find the Remainder for $$3^{100}$$:**

- Divide the exponent $$100$$ by the cycle length (6):
     $$
     100 \div 6 = 16 \, 	ext{remainder} \, 4
     $$
   - Thus:
     $$
     3^{100} \equiv 3^4 \pmod{7}
     $$

- From the cycle:
     $$
     3^4 \mod 7 = 4
     $$

### Final Answer:

The remainder when $$10^{100}$$ is divided by $$7$$ is **4**.  
Correct option: **D**

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