Question

Find the remainder when \(N = 1821 \times 1823 \times 1827\) is divided by 12.

• A. 9
• B. 12
• C. 15
• D. 18

Hint:

Find each number's remainder on dividing by 12 separately, then multiply those three remainders together and reduce mod 12 again, rather than multiplying the full numbers out.

Answer 1

The Correct Option is A

Step 1: Understand what is being asked.
We need the remainder when \(N = 1821 \times 1823 \times 1827\) is divided by 12. Instead of multiplying out the huge number \(N\) first, we can just track each factor's remainder on division by 12 and multiply those remainders.

Step 2: Find each factor's remainder mod 12.
\(1821 = 12 \times 151 + 9\), so \(1821\) leaves remainder \(9\).
\(1823 = 12 \times 151 + 11\), so \(1823\) leaves remainder \(11\).
\(1827 = 12 \times 152 + 3\), so \(1827\) leaves remainder \(3\).

Step 3: Multiply the remainders and reduce again.
\[ 9 \times 11 \times 3 = 297 \]
Now find \(297\) mod 12: \(297 = 12 \times 24 + 9\), so \(297\) leaves remainder \(9\).

Step 4: Conclude.
Since remainders multiply and reduce the same way the original numbers do, \(N\) leaves the same remainder \(9\) when divided by 12.

Final Answer:
The remainder is 9. \[ \boxed{9} \]