`(1001xx1002xx1003)/(27)` find R.

Answer

To find the remainder \( R \) when \( \frac{1001 \times 1002 \times 1003}{27} \), we can follow these steps:

### Step 1: Find the remainders of each number when divided by 27.

1. **Calculate \( 1001 \mod 27 \)**:
- Divide \( 1001 \) by \( 27 \):
\[ 1001 \div 27 \approx 37.037 \quad \text{(take the integer part, which is 37)} \]
- Multiply \( 27 \) by \( 37 \):
\[ 27 \times 37 = 999 \]
- Subtract from \( 1001 \):
\[ 1001 - 999 = 2 \]
- Thus, \( 1001 \mod 27 = 2 \).

2. **Calculate \( 1002 \mod 27 \)**:
- Divide \( 1002 \) by \( 27 \):
\[ 1002 \div 27 \approx 37.067 \quad \text{(take the integer part, which is 37)} \]
- Multiply \( 27 \) by \( 37 \):
\[ 27 \times 37 = 999 \]
- Subtract from \( 1002 \):
\[ 1002 - 999 = 3 \]
- Thus, \( 1002 \mod 27 = 3 \).

3. **Calculate \( 1003 \mod 27 \)**:
- Divide \( 1003 \) by \( 27 \):
\[ 1003 \div 27 \approx 37.096 \quad \text{(take the integer part, which is 37)} \]
- Multiply \( 27 \) by \( 37 \):
\[ 27 \times 37 = 999 \]
- Subtract from \( 1003 \):
\[ 1003 - 999 = 4 \]
- Thus, \( 1003 \mod 27 = 4 \).

### Step 2: Multiply the remainders.

Now we have:
- \( 1001 \mod 27 = 2 \)
- \( 1002 \mod 27 = 3 \)
- \( 1003 \mod 27 = 4 \)

Now, multiply these remainders:
\[ R = (2 \times 3 \times 4) \mod 27 \]
Calculating:
\[ 2 \times 3 = 6 \]
\[ 6 \times 4 = 24 \]

### Step 3: Find the remainder of the product when divided by 27.

Now we need to find \( 24 \mod 27 \):
\[ 24 \div 27 = 0 \quad \text{(the integer part is 0)} \]
Thus, the remainder is simply \( 24 \).

### Final Answer:
The remainder \( R \) when \( \frac{1001 \times 1002 \times 1003}{27} \) is:
\[ \boxed{24} \]