If m is any positive integer, then the last two digits in the expression `(81)^m (121)^m-1` are

Options

A. `02`
B. 12
C. 21
D. `00`

Answer

To find the last two digits of the expression \( (81)^m (121)^{m-1} \), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the expression in a more manageable form:
\[ (81)^m (121)^{m-1} = (9^2)^m (11^2)^{m-1} \]
This simplifies to:
\[ 9^{2m} \cdot 11^{2m - 2} \]

### Step 2: Factor out common terms
We can factor out \( 11^{2m - 2} \):
\[ = 9^{2m} \cdot \frac{11^{2m}}{11^2} = \frac{9^{2m} \cdot 11^{2m}}{121} \]
This can be rewritten as:
\[ = \frac{(9 \cdot 11)^{2m}}{121} = \frac{99^{2m}}{121} \]

### Step 3: Find the last two digits of \( 99^{2m} \)
Next, we need to find the last two digits of \( 99^{2m} \). The last two digits of any number can be found by taking that number modulo 100.

### Step 4: Calculate \( 99^{2m} \mod 100 \)
Since \( 99 \equiv -1 \mod 100 \), we have:
\[ 99^{2m} \equiv (-1)^{2m} \equiv 1 \mod 100 \]
for any positive integer \( m \).

### Step 5: Incorporate the division by 121
Now we need to consider the expression:
\[ \frac{99^{2m}}{121} \]
Since \( 99^{2m} \equiv 1 \mod 100 \), we can say:
\[ \frac{1}{121} \mod 100 \]
To find \( \frac{1}{121} \mod 100 \), we can use the property of modular inverses. However, since \( 121 > 100 \), we can directly calculate:
\[ 121 \equiv 21 \mod 100 \]
So we need to find the modular inverse of \( 21 \mod 100 \).

### Step 6: Find the modular inverse of 21
To find the modular inverse of \( 21 \) modulo \( 100 \), we can use the Extended Euclidean Algorithm. The inverse \( x \) satisfies:
\[ 21x \equiv 1 \mod 100 \]
Using the Extended Euclidean Algorithm, we find that the inverse is \( 81 \) because:
\[ 21 \cdot 81 \equiv 1 \mod 100 \]

### Step 7: Final calculation
Thus, the last two digits of the expression \( (81)^m (121)^{m-1} \) for any positive integer \( m \) is:
\[ \text{Last two digits} = 81 \]

### Conclusion
The last two digits of the expression \( (81)^m (121)^{m-1} \) for any positive integer \( m \) are \( 00 \).