Question

$N = 1421 \times 1423 \times 1425$. What is the remainder when $N$ is divided by $12$?

• A. 0
• B. 9
• C. 3
• D. 6

Hint:

When finding a remainder for a large product, reduce each factor modulo $m$ first, then multiply.

Answer 1

The Correct Option is C

We can find the remainder of $N$ modulo 12 without full multiplication. Step 1: Find each factor mod 12 \[ 1421 \div 12 = 118 \text{ remainder } 5 \quad \Rightarrow \quad 1421 \equiv 5 \ (\text{mod } 12) \] \[ 1423 \div 12 = 118 \text{ remainder } 7 \quad \Rightarrow \quad 1423 \equiv 7 \ (\text{mod } 12) \] \[ 1425 \div 12 = 118 \text{ remainder } 9 \quad \Rightarrow \quad 1425 \equiv 9 \ (\text{mod } 12) \] Step 2: Multiply mod 12 First two: $5 \times 7 = 35 \equiv 11 \ (\text{mod } 12)$ Multiply by 9: $11 \times 9 = 99 \equiv 3 \ (\text{mod } 12)$. \[ \boxed{\text{Remainder} = 3} \]