Write the remainder obtained when 1! + 2! + 3! + ... + 200! is di… - Vidyadip

Question

Write the remainder obtained when 1! + 2! + 3! + ... + 200! is divided by 14.

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Answer

First we will find the least factorial term divisible by 14.
As 7!=7 × 6 × 5! is divisible by 14 leaving remainder zero.
Hence terms 7! onwards can be written as multiple of 7!.
8!=8 × 7!, 9! = 9 × 8 × 7!... like ways 200! can also be written as multiple of 7!.
So all the terms 7! onwards are divisible by 14 leaving remainder zero.
= 1! + 2! + 3! + 4! + 5! + 6!
= 1 + 2 + 6 + 24 + 120 + 720
= 873
Hence remainder obtained when 1! + 2! + 3! + ….+ 200! is divided by 14 is 5.

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