Find the last digit, last two digits and last three digits of the number `(81)^(25)`

Answer

To find the last digit, last two digits, and last three digits of \( (81)^{25} \), we can follow these steps:

### Step 1: Rewrite the base
First, we can express \( 81 \) as \( 9^2 \):
\[ (81)^{25} = (9^2)^{25} = 9^{50} \]

### Step 2: Use Binomial Theorem
Next, we can express \( 9 \) as \( 10 - 1 \):
\[ 9^{50} = (10 - 1)^{50} \]
Now, we will use the Binomial Theorem to expand \( (10 - 1)^{50} \):
\[ (10 - 1)^{50} = \sum_{r=0}^{50} \binom{50}{r} 10^{50-r} (-1)^r \]

### Step 3: Identify relevant terms for last digits
To find the last digit, last two digits, and last three digits, we only need the last three terms of the expansion:
1. The last term (when \( r = 50 \)):
\[ \binom{50}{50} 10^{0} (-1)^{50} = 1 \]
2. The second last term (when \( r = 49 \)):
\[ \binom{50}{49} 10^{1} (-1)^{49} = 50 \cdot 10 \cdot (-1) = -500 \]
3. The third last term (when \( r = 48 \)):
\[ \binom{50}{48} 10^{2} (-1)^{48} = \binom{50}{2} 10^{2} = \frac{50 \times 49}{2} \cdot 100 = 1225 \cdot 100 = 122500 \]

### Step 4: Combine the terms
Now, we combine these three terms:
\[ (10 - 1)^{50} = 122500 - 500 + 1 = 122001 \]

### Step 5: Extract last digits
Now we can extract the last digit, last two digits, and last three digits from \( 122001 \):
- Last digit: \( 1 \)
- Last two digits: \( 01 \)
- Last three digits: \( 001 \)

### Final Answer
Thus, the last digit, last two digits, and last three digits of \( (81)^{25} \) are:
- Last digit: \( 1 \)
- Last two digits: \( 01 \)
- Last three digits: \( 001 \)