If $10^{68}$ is divided by 13 , the remainder is

Question

Accepted Answer

To find the remainder when $$10^{68}$$ is divided by 13, we use modular arithmetic and observe the cyclic nature of remainders.

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### Step 1: Compute powers of 10 modulo 13  
Calculate the remainders for successive powers of $$10$$ modulo $$13$$:  
- $$10^1 \mod 13 = 10$$  
- $$10^2 \mod 13 = 100 \mod 13 = 9$$  
- $$10^3 \mod 13 = 1000 \mod 13 = 12$$  
- $$10^4 \mod 13 = 10000 \mod 13 = 3$$  
- $$10^5 \mod 13 = 30000 \mod 13 = 4$$  
- $$10^6 \mod 13 = 400000 \mod 13 = 1$$

The remainders repeat every cycle of six powers: $$10, 9, 12, 3, 4, 1$$.

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### Step 2: Determine the position of $$10^{68}$$ in the cycle  
Since the remainders repeat every six powers, divide the exponent $$68$$ by $$6$$:  
- $$68 \div 6 = 11$$ remainder $$2$$.

This means $$10^{68} \equiv 10^2 \mod 13$$.

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### Step 3: Find the remainder for $$10^2$$ modulo $$13$$  
From Step 1, we know:  
$$10^2 \mod 13 = 9$$.

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### Final Answer:
The remainder when $$10^{68}$$ is divided by $$13$$ is **9**.

**Correct Option:** B

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