`(10^(1)+10^(2)+10^(3)------+10^(100))/(6)` find R.

Answer

To solve the problem \((10^1 + 10^2 + 10^3 + \ldots + 10^{100}) / 6\) and find the remainder \(R\), we can follow these steps:

### Step 1: Identify the series
The expression \(10^1 + 10^2 + 10^3 + \ldots + 10^{100}\) is a geometric series where the first term \(a = 10\) and the common ratio \(r = 10\).

### Step 2: Use the formula for the sum of a geometric series
The sum \(S_n\) of the first \(n\) terms of a geometric series can be calculated using the formula:
\[ S_n = a \frac{(r^n - 1)}{(r - 1)} \]
In our case, \(n = 100\), \(a = 10\), and \(r = 10\):
\[ S_{100} = 10 \frac{(10^{100} - 1)}{(10 - 1)} = 10 \frac{(10^{100} - 1)}{9} \]

### Step 3: Simplify the sum
Now we have:
\[ S_{100} = \frac{10^{101} - 10}{9} \]

### Step 4: Divide the sum by 6
We need to find:
\[ \frac{S_{100}}{6} = \frac{10^{101} - 10}{54} \]

### Step 5: Find the remainder when divided by 6
To find the remainder when \(S_{100}\) is divided by 6, we can find the remainder of \(10^{101} - 10\) when divided by 54.

### Step 6: Calculate \(10^n \mod 6\)
Notice that:
- \(10 \mod 6 = 4\)
- \(10^2 \mod 6 = 4^2 \mod 6 = 16 \mod 6 = 4\)
- Continuing this, we see that \(10^n \mod 6 = 4\) for any \(n \geq 1\).

Thus:
\[ 10^{101} \mod 6 = 4 \]
and
\[ 10 \mod 6 = 4 \]

### Step 7: Combine the results
Now we can calculate:
\[ 10^{101} - 10 \mod 6 = 4 - 4 = 0 \]

### Step 8: Find the remainder when divided by 6
Since \(10^{101} - 10 \equiv 0 \mod 6\), we have:
\[ \frac{10^{101} - 10}{54} \mod 6 = 0 \]

### Final Answer
Thus, the remainder \(R\) when \((10^1 + 10^2 + 10^3 + \ldots + 10^{100})\) is divided by 6 is:
\[ \boxed{0} \]