Question

The remainder obtained when \(1! + 2! + \cdots + 200!\) is divided by 14 is

• A. 3
• B. 4
• C. 5
• D. None of the above

Hint:

For \(n \ge 7\), \(n!\) is divisible by 14.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For \(n \ge 7\), \(n!\) is divisible by 14.
Step 2: Detailed Explanation:
\(14 = 2 \times 7\)
For \(n \ge 7\), \(n!\) contains both 2 and 7, so divisible by 14.
So only need sum \(1! + 2! + 3! + 4! + 5! + 6!\)
= \(1 + 2 + 6 + 24 + 120 + 720 = 873\)
\(873 \div 14\): \(14 \times 62 = 868\), remainder 5
Step 3: Final Answer:
Remainder = 5.