What is the remainder obtained when 1421 × 1423 × 1425 is divided by 12 ?

Question

Accepted Answer

To determine the remainder when the product 1421 × 1423 × 1425 is divided by 12, we can use modular arithmetic to simplify the calculations.

First, let's find the remainder of each number when divided by 12:

1. **1421 mod 12:**
   - 1421 divided by 12 gives a quotient of 118 and a remainder of 5 (since 1421 = 12 × 118 + 5).
   - Therefore, 1421 ≡ 5 (mod 12).

2. **1423 mod 12:**
   - 1423 divided by 12 gives a quotient of 118 and a remainder of 7 (since 1423 = 12 × 118 + 7).
   - Therefore, 1423 ≡ 7 (mod 12).

3. **1425 mod 12:**
   - 1425 divided by 12 gives a quotient of 118 and a remainder of 9 (since 1425 = 12 × 118 + 9).
   - Therefore, 1425 ≡ 9 (mod 12).

Now, we need to find the remainder of the product 5 × 7 × 9 when divided by 12:

- Calculate the product: 5 × 7 = 35.
- Find 35 mod 12: 35 divided by 12 gives a remainder of 11 (since 35 = 12 × 2 + 11).
- Therefore, 5 × 7 ≡ 11 (mod 12).

Next, multiply this result by 9:

- Calculate the product: 11 × 9 = 99.
- Find 99 mod 12: 99 divided by 12 gives a remainder of 3 (since 99 = 12 × 8 + 3).
- Therefore, 11 × 9 ≡ 3 (mod 12).

Thus, the remainder when 1421 × 1423 × 1425 is divided by 12 is 3.

**Explanation of why other options are incorrect:**

- **Option 1 (Remainder 1):** The calculations show that the remainder is 3, not 1. Therefore, Option 1 is incorrect.
- **Option 2 (Remainder 2):** The calculations show that the remainder is 3, not 2. Therefore, Option 2 is incorrect.
- **Option 4 (Remainder 4):** The calculations show that the remainder is 3, not 4. Therefore, Option 4 is incorrect.

In conclusion, the correct answer is Option 3, as the remainder when 1421 × 1423 × 1425 is divided by 12 is indeed 3.