Question

Find the remainder when: (1719 × 1721 × 1723 × 1725 × 1727) is divided by 18?

• A. 9
• B. 7
• C. 19
• D. 3
• E. 7

Hint:

Use modulo arithmetic to simplify large products; reduce each factor modulo the divisor.

Answer 1

The Correct Option is A

Step 1: Using Modulo Arithmetic:
Find each number modulo 18.
1719 ÷ 18: 18 × 95 = 1710, remainder 9. So 1719 ≡ 9 (mod 18)
1721 ≡ 9 + 2 = 11 (mod 18)
1723 ≡ 11 + 2 = 13 (mod 18)
1725 ≡ 13 + 2 = 15 (mod 18)
1727 ≡ 15 + 2 = 17 (mod 18)

Step 2: Product Modulo 18:
Product ≡ \(9 \times 11 \times 13 \times 15 \times 17\) (mod 18)
Simplify step by step:
\(9 \times 11 = 99 ≡ 99 - 5×18 = 99 - 90 = 9\) (mod 18)
\(9 \times 13 = 117 ≡ 117 - 6×18 = 117 - 108 = 9\) (mod 18)
\(9 \times 15 = 135 ≡ 135 - 7×18 = 135 - 126 = 9\) (mod 18)
\(9 \times 17 = 153 ≡ 153 - 8×18 = 153 - 144 = 9\) (mod 18)
Thus, remainder = 9.