Question

Remainder when (1! + 2! + 3! + . . . . + 4000!) is divided by 7

Answer 1

the remainder should be 2
so that it will get8
8 canbe divided by 40000

Answer 2

SOLUTION

TO CHOOSE THE CORRECT OPTION

Remainder when ( 1! + 2! + 3! + ... + 4000! ) is divided by 7

a) 7

b) 1

c) 5

d) None

EVALUATION

Here the given expression is

1 ! + 2 ! + 3 ! + 4 ! + 5 ! + 6 ! + 7 ! +... + 4000 !

We see that

Starting from 7 ! upto 4000 ! each terms contain the number 7

So each of 7 ! , 8 ! , 9 ! , ... , 4000 ! is divisible by 7

Now we have to check the expression

1 ! + 2 ! + 3 ! + 4 ! + 5 ! + 6 !

Here

5 ! + 6 ! = ( 1 + 6 ) 5 ! = 7 × 5 !

So 5 ! + 6 ! is divisible by 7

1 ! + 3 ! = 1 + 6 = 7

So 1 ! + 3 ! is divisible by 7

Now

2 ! + 4 ! = 2 + 24 = 26

26 = ( 3 × 7 ) + 5

By Division algorithm the Remainder when 26 is divided by 7 is 5

Hence the required Remainder when ( 1! + 2! + 3! + ... + 4000! ) is divided by 7 is 5

FINAL ANSWER

Hence the correct option is c) 5

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