Question

please answer this its very urgent​

Answer 1

Let us see this.

We know that 4! =24 which is divisible by 4.

Now, after 4!,we can write each factorial term with as 4! * y where y is a positive number.

ex :- 5! = 4! *5

6! = 4! * 5 * 6.

So 4! is a factor of 5!,6!,7!, ... ,100!.

So , from 4! to 100!, each term is divisible by 4!(24) as they have 4! as their factor.

So, 4! mod 24 = 0.

Similarly , 5! %24 =0 and so for rest upto 100!.

Now, Rule of modular summation states that

(a+b)%c = (a%c + b%c) %c.

Now, (1! +2! +3! +4! ...+100!) mod 24 =

(1!mod24 + 2!mod24 + 3!mod 24 + 4! mod24 + 5!mod24 + 6!mod24 +... + 100!mod24)%24

=( 1 mod 24 + 2 mod 24 + 6 mod 24 + 0 + 0 +0 .. +0)mod 24

= (1 + 2 +6) mod 24

= 9 mod 24

=9.

Hence , 9 is the required answer.

Hope that helps. :)