Question:
The remainder obtained when \(1! + 2! + \cdots + 200!\) is divided by 14 is
The remainder obtained when \(1! + 2! + \cdots + 200!\) is divided by 14 is
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For \(n \ge 7\), \(n!\) is divisible by 14.
Updated On: Apr 7, 2026
- 3
- 4
- 5
- None of the above
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The Correct Option is C
Solution and Explanation
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.
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.
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