What is the remainder when `(10 + 10^2 + 10^3 + 10^4 + 10^5)` is divided by 6 ?

Answer

To find the remainder when \(10 + 10^2 + 10^3 + 10^4 + 10^5\) is divided by 6, we can follow these steps:

### Step 1: Calculate each power of 10 modulo 6
First, we need to find the value of \(10\) modulo \(6\):
\[ 10 \mod 6 = 4 \]
Thus, we can replace \(10\) with \(4\) in our expression.

### Step 2: Rewrite the expression using the modulo result
Now, we can rewrite the entire expression:
\[ 10 + 10^2 + 10^3 + 10^4 + 10^5 \equiv 4 + 4^2 + 4^3 + 4^4 + 4^5 \mod 6 \]

### Step 3: Calculate each power of 4 modulo 6
Next, we calculate \(4^2\), \(4^3\), \(4^4\), and \(4^5\) modulo \(6\):
- \(4^1 \mod 6 = 4\)
- \(4^2 = 16 \mod 6 = 4\)
- \(4^3 = 64 \mod 6 = 4\)
- \(4^4 = 256 \mod 6 = 4\)
- \(4^5 = 1024 \mod 6 = 4\)

### Step 4: Sum the results
Now we can sum these results:
\[ 4 + 4 + 4 + 4 + 4 = 20 \]

### Step 5: Find the remainder when the sum is divided by 6
Now we need to find the remainder of \(20\) when divided by \(6\):
\[ 20 \mod 6 = 2 \]

### Conclusion
Thus, the remainder when \(10 + 10^2 + 10^3 + 10^4 + 10^5\) is divided by \(6\) is:
\[ \boxed{2} \]