`(15^(40))/(100)` find R.

Answer

To find the remainder of \( \frac{15^{40}}{100} \), we can follow these steps:

### Step 1: Factor the denominator
We can express \( 100 \) as \( 10^2 \) or \( 2^2 \times 5^2 \). This will help us apply the Chinese Remainder Theorem later.

### Step 2: Calculate \( 15^{40} \mod 4 \)
First, we find \( 15 \mod 4 \):
\[ 15 \equiv 3 \mod 4 \]
Now, we calculate \( 15^{40} \mod 4 \):
\[ 15^{40} \equiv 3^{40} \mod 4 \]
Since \( 3^2 \equiv 1 \mod 4 \), we can simplify:
\[ 3^{40} = (3^2)^{20} \equiv 1^{20} \equiv 1 \mod 4 \]

### Step 3: Calculate \( 15^{40} \mod 25 \)
Next, we find \( 15 \mod 25 \):
\[ 15 \equiv 15 \mod 25 \]
Now, we calculate \( 15^{40} \mod 25 \) using Euler's theorem. First, we find \( \phi(25) \):
\[ \phi(25) = 25 \left(1 - \frac{1}{5}\right) = 25 \cdot \frac{4}{5} = 20 \]
Since \( 15 \) and \( 25 \) are coprime, we can use Euler's theorem:
\[ 15^{20} \equiv 1 \mod 25 \]
Thus,
\[ 15^{40} = (15^{20})^2 \equiv 1^2 \equiv 1 \mod 25 \]

### Step 4: Combine results using the Chinese Remainder Theorem
Now we have:
\[ 15^{40} \equiv 1 \mod 4 \]
\[ 15^{40} \equiv 1 \mod 25 \]
Since both congruences yield the same result, we can conclude:
\[ 15^{40} \equiv 1 \mod 100 \]

### Step 5: Find the remainder
Now, we need to find the remainder of \( 15^{40} \) divided by \( 100 \):
\[ \frac{15^{40}}{100} \text{ gives a remainder of } 1. \]

### Final Calculation
Since we need the remainder when \( 15^{40} \) is divided by \( 100 \):
\[ \text{Remainder } R = 1. \]

### Conclusion
Thus, the remainder \( R \) when \( 15^{40} \) is divided by \( 100 \) is:
\[ \boxed{1} \]