The greatest integer which divides the number `(101)^(100)-1` is equal to

Options

A. `10^(2)`
B. `10^(3)`
C. `10^(4)`
D. `10^(5)`

Answer

To find the greatest integer that divides \( (101)^{100} - 1 \), we can use the Binomial Theorem and properties of divisibility. Let's break it down step by step:

### Step 1: Rewrite the Expression
We start with the expression \( (101)^{100} - 1 \). We can use the Binomial Theorem to expand \( (100 + 1)^{100} \):
\[ (101)^{100} = (100 + 1)^{100} \]

### Step 2: Apply the Binomial Theorem
Using the Binomial Theorem, we can expand \( (100 + 1)^{100} \):
\[ (100 + 1)^{100} = \sum_{k=0}^{100} \binom{100}{k} 100^{100-k} \cdot 1^k \]
This gives us:
\[ (100 + 1)^{100} = \binom{100}{0} 100^{100} + \binom{100}{1} 100^{99} + \binom{100}{2} 100^{98} + \ldots + \binom{100}{100} \]

### Step 3: Subtract 1
Now, we need to subtract 1 from the expansion:
\[ (101)^{100} - 1 = \left( \binom{100}{0} 100^{100} + \binom{100}{1} 100^{99} + \ldots + \binom{100}{100} \right) - 1 \]
The last term \( \binom{100}{100} \) is 1, so we have:
\[ (101)^{100} - 1 = \binom{100}{0} 100^{100} + \binom{100}{1} 100^{99} + \ldots + \binom{100}{99} 100 + (0) \]

### Step 4: Factor Out Common Terms
Notice that all terms except the last one are multiples of \( 100 \):
\[ (101)^{100} - 1 = 100 \left( \binom{100}{0} 100^{99} + \binom{100}{1} 100^{98} + \ldots + \binom{100}{99} \right) \]
This shows that \( (101)^{100} - 1 \) is divisible by \( 100 \).

### Step 5: Determine the Highest Power of 10
Next, we need to find the highest power of 10 that divides \( (101)^{100} - 1 \). Since \( 100 = 10^2 \), we can conclude that \( (101)^{100} - 1 \) is divisible by \( 10^2 \).

### Step 6: Check for Higher Powers
To check for higher powers of 10, we can look at the factors of \( 100 \):
\[ 100 = 2^2 \cdot 5^2 \]
Thus, \( (101)^{100} - 1 \) is divisible by \( 10^2 \), but we need to check if it is divisible by \( 10^3 \) or \( 10^4 \).

### Step 7: Conclusion
After examining the factors, we find that the greatest integer that divides \( (101)^{100} - 1 \) is:
\[ \boxed{10000} \]