The number obtained after dividing the number formed by the last three digits of `17^(256)` by 100 is

Answer

To find the number obtained after dividing the number formed by the last three digits of \( 17^{256} \) by 100, we will follow these steps:

### Step 1: Find the last three digits of \( 17^{256} \)

To find the last three digits of \( 17^{256} \), we can compute \( 17^{256} \mod 1000 \).

### Step 2: Use the property of powers

We can express \( 17^{256} \) as \( (17^4)^{64} \). First, we need to calculate \( 17^4 \).

\[ 17^2 = 289 \]
\[ 17^4 = 289^2 = 83521 \]

Now, we find \( 83521 \mod 1000 \):

\[ 83521 \mod 1000 = 521 \]

So, \( 17^4 \equiv 521 \mod 1000 \).

### Step 3: Calculate \( 17^{256} \)

Now, we calculate \( (17^4)^{64} \mod 1000 \):

\[ 17^{256} \equiv 521^{64} \mod 1000 \]

### Step 4: Use binomial expansion

To simplify \( 521^{64} \), we can express it as \( (520 + 1)^{64} \) and use the binomial theorem:

\[ (520 + 1)^{64} = \sum_{k=0}^{64} \binom{64}{k} \cdot 520^k \cdot 1^{64-k} \]

### Step 5: Calculate significant terms

We only need the last three digits, so we can ignore terms with \( k \geq 3 \) since they will contribute to higher powers of 1000.

Calculating the first few terms:

- For \( k = 0 \):
\[ \binom{64}{0} \cdot 520^0 \cdot 1^{64} = 1 \]

- For \( k = 1 \):
\[ \binom{64}{1} \cdot 520^1 \cdot 1^{63} = 64 \cdot 520 = 33280 \]

- For \( k = 2 \):
\[ \binom{64}{2} \cdot 520^2 \cdot 1^{62} = \frac{64 \cdot 63}{2} \cdot 270400 = 20172800 \]

Now, we only need the last three digits of these sums:

- \( 1 \mod 1000 = 1 \)
- \( 33280 \mod 1000 = 280 \)
- \( 20172800 \mod 1000 = 800 \)

### Step 6: Sum the contributions

Now, we sum these contributions:

\[ 1 + 280 + 800 = 1081 \]

### Step 7: Find the last three digits

Now, we take \( 1081 \mod 1000 \):

\[ 1081 \mod 1000 = 81 \]

### Step 8: Divide by 100

Finally, we divide the last three digits by 100:

\[ \frac{81}{100} = 0.81 \]

Thus, the number obtained after dividing the number formed by the last three digits of \( 17^{256} \) by 100 is:

\[ \boxed{0.81} \]