If `10^m` divides the number `101^(100)-1` then, find the greatest value of `mdot`

Explanation

To solve the problem of finding the greatest value of \( m \) such that \( 10^m \) divides \( 101^{100} - 1 \), we can follow these steps:

### Step 1: Understand the expression
We start with the expression \( 101^{100} - 1 \). We can rewrite \( 101 \) as \( 100 + 1 \).

### Step 2: Use the Binomial Theorem
Using the Binomial Theorem, we can expand \( (100 + 1)^{100} \):
\[ (100 + 1)^{100} = \sum_{k=0}^{100} \binom{100}{k} 100^k \cdot 1^{100-k} \]
This expands to:
\[ = \binom{100}{0} 100^0 + \binom{100}{1} 100^1 + \binom{100}{2} 100^2 + \ldots + \binom{100}{100} 100^{100} \]
\[ = 1 + 100 \cdot 100 + \frac{100 \cdot 99}{2} \cdot 100^2 + \ldots + 100^{100} \]

### Step 3: Subtract 1
Now, we subtract 1 from the expansion:
\[ 101^{100} - 1 = 100^2 + \binom{100}{2} 100^2 + \ldots + 100^{100} \]
This means:
\[ 101^{100} - 1 = 100^2 \left(1 + \binom{100}{2} 100^{0} + \ldots + 100^{98}\right) \]

### Step 4: Factor out \( 100^2 \)
The term \( 100^2 \) can be expressed as \( (10^2)^2 = 10^4 \). Thus, we have:
\[ 101^{100} - 1 = 10^4 \cdot \left(1 + \text{other terms}\right) \]

### Step 5: Determine the highest power of 10
To find the greatest value of \( m \) such that \( 10^m \) divides \( 101^{100} - 1 \), we note that \( 10^4 \) is the factor we have extracted. The other terms do not contribute any additional factors of 10.

### Conclusion
Thus, the greatest value of \( m \) such that \( 10^m \) divides \( 101^{100} - 1 \) is:
\[ \boxed{4} \]