Find the remainder in expression–
`(1235 xx 1237 xx 1239)/(12)`

Options

A. 3
B. 6
C. 11
D. 9

Answer

To find the remainder of the expression \( \frac{1235 \times 1237 \times 1239}{12} \), we can follow these steps:

### Step 1: Find the remainder of each number when divided by 12.

1. **Calculate \( 1235 \mod 12 \)**:
- Divide 1235 by 12:
\[ 1235 \div 12 = 102.9167 \quad \text{(take the integer part, which is 102)} \]
- Multiply 102 by 12:
\[ 102 \times 12 = 1224 \]
- Subtract from 1235 to find the remainder:
\[ 1235 - 1224 = 11 \]
- So, \( 1235 \mod 12 = 11 \).

2. **Calculate \( 1237 \mod 12 \)**:
- Divide 1237 by 12:
\[ 1237 \div 12 = 103.0833 \quad \text{(take the integer part, which is 103)} \]
- Multiply 103 by 12:
\[ 103 \times 12 = 1236 \]
- Subtract from 1237 to find the remainder:
\[ 1237 - 1236 = 1 \]
- So, \( 1237 \mod 12 = 1 \).

3. **Calculate \( 1239 \mod 12 \)**:
- Divide 1239 by 12:
\[ 1239 \div 12 = 103.25 \quad \text{(take the integer part, which is 103)} \]
- Multiply 103 by 12:
\[ 103 \times 12 = 1236 \]
- Subtract from 1239 to find the remainder:
\[ 1239 - 1236 = 3 \]
- So, \( 1239 \mod 12 = 3 \).

### Step 2: Multiply the remainders and find the remainder of the product when divided by 12.

Now we have:
- \( 1235 \mod 12 = 11 \)
- \( 1237 \mod 12 = 1 \)
- \( 1239 \mod 12 = 3 \)

Next, we calculate:
\[ (11 \times 1 \times 3) \mod 12 \]

Calculating the product:
\[ 11 \times 1 = 11 \]
\[ 11 \times 3 = 33 \]

Now, find \( 33 \mod 12 \):
- Divide 33 by 12:
\[ 33 \div 12 = 2.75 \quad \text{(take the integer part, which is 2)} \]
- Multiply 2 by 12:
\[ 2 \times 12 = 24 \]
- Subtract from 33 to find the remainder:
\[ 33 - 24 = 9 \]

### Final Answer:
The remainder when \( 1235 \times 1237 \times 1239 \) is divided by 12 is **9**.

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