Question

Show that out of every four consecutive positive integer one will be divisible by four.

Answer 1

Proof by contradiction:

Let the 4 nos be a,a+1,a+2,a+3 - none of which are divisible by 4

Nos not divisible by 4 are either 4k+1,4k+2 or 4k+3, where k is a whole number

Case 1: a is 4k+1

Now, a+3 becomes 4k+4 which is divisible by 4 - so this case is not possible

Case 2: a is 4k+2

Now, a+2 becomes 4k+4 which is divisible by 4 - so this case is not possible

Case 3: a is 4k+3

Now, a+1 becomes 4k+4 which is divisible by 4 - so this case is also not possible

We have exhausted all possible cases, and in all cases non-divisibility by 4 is impossible.

Hence proved by contradiction

Answer 2

It is due to this reason that afeter every 4 no we do have a multiple of 4 like in

1,2,3,4.we do have 4 which is 1*4