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: Spot a multiple of 12 close to all three numbers.
Notice that $1824 = 12 \times 152$ is exactly divisible by 12, and all three factors sit close to it: $1821 = 1824 - 3$, $1823 = 1824 - 1$, and $1827 = 1824 + 3$.

Step 2: Replace each factor with its distance from 1824.
Since $1824$ contributes nothing to the remainder (it is a clean multiple of 12), each factor behaves, for remainder purposes, just like its small offset:
\[ 1821 \equiv -3, \quad 1823 \equiv -1, \quad 1827 \equiv 3 \pmod{12} \]

Step 3: Multiply the offsets.
\[ (-3) \times (-1) \times 3 = 9 \]
This is already between 0 and 11, so no further reduction is needed.

Step 4: Double check by converting the negative remainders to positive form.
$-3 \equiv 9 \pmod{12}$ and $-1 \equiv 11 \pmod{12}$, which matches the direct-division remainders 9 and 11 found by long division, confirming the same product $9 \times 11 \times 3 = 297$, which reduces to 9 mod 12.

Final Answer:
$N$ leaves remainder 9 when divided by 12. \[ \boxed{9} \]