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

The problem requires us to find the remainder when the sum \(1! + 2! + \cdots + 200!\) is divided by 14. To solve this, let's evaluate the remainder of each factorial individually and then sum them up. 

1. The factorial of a number \(n!\) for \(n \geq 5\) is divisible by both 7 and 2, and hence by 14. Therefore, for \(5! \) and above, the remainder when divided by 14 is 0.
2. We need to consider only \(1! \) to \(4! \), as these could potentially contribute a non-zero remainder:
   • \(1! = 1\). The remainder when \(1\) is divided by 14 is 1.
   • \(2! = 2\). The remainder when \(2\) is divided by 14 is 2.
   • \(3! = 6\). The remainder when \(6\) is divided by 14 is 6.
   • \(4! = 24\). The remainder when \(24\) is divided by 14 is 10.
3. Adding up these remainders: \(1 + 2 + 6 + 10 = 19\).
4. Now, we find the remainder when 19 is divided by 14: \(19 \div 14 = 1\) with a remainder of 5.

Therefore, the remainder obtained when \(1! + 2! + \cdots + 200!\) is divided by 14 is \(5\).

Thus, the correct answer is 5.