Reasoning ProofSolution1 100 is an even number is TRUE 100

Reasoning & Proof

Solution

1. 100! is an even number is TRUE

100! = 100 X 99 X 98 X 97 X 96 X ..........X 2 X 1

In this we have the whole multiplication multiplied by 2

As we know, any number when either multiplied or exactly divisible by 2 is an even number and hence in this case 2 is being multiplied and hence 100! is an even number  

2. 100! is divisible by 101 is FALSE

     101 is a prime number

      100! = -1 mod 101

      100! + 1 = 0 mod 101

       It would have been divisible if 1 is added to 100! and hence in this case 100! is not divisible by 101

3. 100! is larger than 101! - 100! is FALSE

     101! - 100!

      ( 101 X 100! ) - 100!

     100! (101 - 1)  

     100! X 100 > 100!

The above equation is anyhow greater than 100! since 100 is multiplied to 100! and hence the given equation

100! > 101! - 100! is FALSE

4. 100! is larger than 50! X 10⁵⁰   is TRUE

     100! > 50! X 10⁵⁰

       100 X 99 X 98 X 97 ........52 X 51X 50! > 50! X 10⁵⁰

       50! on both sides get cancelled

     100 X 99 X 98 X 97......X 52 X 51 >   10 (50 times)

     In the above equation, both sides we have 50 numbers but every number on the left hand side is greater than the numbers on the right hand side and hence even their multiplication will be greater    

This proves the above equation and hence the given equation is TRUE