When `(10^10+10^100+10^1000+........+10^10000000000)` is divided by 7, then the remainder is?

Options

A. 1
B. 5
C. 2
D. 3

Answer

To solve the problem of finding the remainder when \( (10^{10} + 10^{100} + 10^{1000} + \ldots + 10^{10000000000}) \) is divided by 7, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Series**: The expression consists of a series of terms where the exponent of 10 increases by a factor of 10 each time. The series can be expressed as:
\[ S = 10^{10} + 10^{100} + 10^{1000} + \ldots + 10^{10000000000} \]

2. **Determine the Number of Terms**: The exponents are \( 10, 100, 1000, \ldots, 10^{10} \). The last term has \( 10^{10} \) which means there are 10 terms in total.

3. **Find the Remainder of Each Term Modulo 7**: We need to find \( 10^n \mod 7 \) for each term:
- First, calculate \( 10 \mod 7 \):
\[ 10 \equiv 3 \mod 7 \]
- Now, we will find the powers of 3 modulo 7:
- \( 3^1 \equiv 3 \mod 7 \)
- \( 3^2 \equiv 9 \equiv 2 \mod 7 \)
- \( 3^3 \equiv 6 \mod 7 \)
- \( 3^4 \equiv 18 \equiv 4 \mod 7 \)
- \( 3^5 \equiv 12 \equiv 5 \mod 7 \)
- \( 3^6 \equiv 15 \equiv 1 \mod 7 \)

The powers of 3 repeat every 6 terms. Therefore, we can find \( 10^n \mod 7 \) by determining \( n \mod 6 \).

4. **Calculate \( n \mod 6 \)** for each exponent:
- For \( n = 10 \):
\[ 10 \mod 6 = 4 \quad \Rightarrow \quad 10^{10} \equiv 3^4 \equiv 4 \mod 7 \]
- For \( n = 100 \):
\[ 100 \mod 6 = 4 \quad \Rightarrow \quad 10^{100} \equiv 3^4 \equiv 4 \mod 7 \]
- For \( n = 1000 \):
\[ 1000 \mod 6 = 4 \quad \Rightarrow \quad 10^{1000} \equiv 3^4 \equiv 4 \mod 7 \]
- This pattern continues for all terms since all \( n \) values (10, 100, 1000, ..., \( 10^{10} \)) give \( n \mod 6 = 4 \).

5. **Sum the Remainders**: Since each term contributes a remainder of 4:
\[ S \equiv 4 + 4 + 4 + \ldots + 4 \quad (\text{10 terms}) \equiv 10 \times 4 = 40 \mod 7 \]

6. **Calculate \( 40 \mod 7 \)**:
\[ 40 \div 7 = 5 \quad \text{remainder } 5 \]
Thus,
\[ 40 \equiv 5 \mod 7 \]

### Final Answer:
The remainder when \( (10^{10} + 10^{100} + 10^{1000} + \ldots + 10^{10000000000}) \) is divided by 7 is **5**.