Show that out of every four consecutive positive integer one will be d

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

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

Answer»

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