Find the remainder in expression–
`(1001 xx 1002 xx 1003 xx 1004)/(27)`

Options

A. 10
B. 11
C. 2
D. 12

Answer

To find the remainder of the expression \( \frac{1001 \times 1002 \times 1003 \times 1004}{27} \), we can follow these steps:

### Step 1: Rewrite the numbers
We can express each number in terms of 1000:
- \( 1001 = 1000 + 1 \)
- \( 1002 = 1000 + 2 \)
- \( 1003 = 1000 + 3 \)
- \( 1004 = 1000 + 4 \)

### Step 2: Calculate the product modulo 27
Instead of calculating the entire product and then dividing, we can find the product modulo 27 directly. We will calculate the remainders of each number when divided by 27:
- \( 1001 \mod 27 = 1 \)
- \( 1002 \mod 27 = 2 \)
- \( 1003 \mod 27 = 3 \)
- \( 1004 \mod 27 = 4 \)

### Step 3: Multiply the remainders
Now we multiply these remainders together:
\[ 1 \times 2 \times 3 \times 4 = 24 \]

### Step 4: Find the remainder of the product modulo 27
Now, we need to find the remainder when 24 is divided by 27:
\[ 24 \mod 27 = 24 \]

### Conclusion
Thus, the remainder when \( 1001 \times 1002 \times 1003 \times 1004 \) is divided by 27 is **24**.