Question

Prove that only one of the numbers n, n+1 or n +2 is divisible by 3, where n is any positive integer.

Explain ?

Answer 1

Step-by-step explanation:

ANSWER

Since n, n+1, n+2 are three consecutive integers then there must be one number divisible by 3 at least.  

If the remainder at dividing n by 3 is 1, then n+2 must be divisible by 3 and if the remainder at dividing n by 3 is 2, then n+1 must be divisible by 3. Similarly for n+1 and n+2.

Let n be divisible by 3.

3

n+1

​  

=  

3

n

​  

+  

3

1

​  

 

Now, n is divisible by 3 but 1 is not. So we get n+1 not divisible by 3. Similarly,n+2 will not be divisible by 3 as well if n is divisible by 3.

3

n+2

​  

=  

3

n

​  

+  

3

2

​  

 

In the same way, if n+1 is divisible by 3 then n and n+2 can't be divisible by 3. If n+2 is divisible by 3 then n and n+1 cannot be divisible by 3.