Question

Show that exactly one of the numbers n, n + 2 or n + 4 is divisible by 3

Answer 1

Here is your answer
We know that any positive integer is of the form 3q,3q+1,3q+2 for some integer q 

    CASE 1

When n =3q , is divisible by 3 

n+2= 3q+2 , is not divisible by 3 as it leaves a remainder 2

n+4 = 3q+4, 3q + 3 +1, 3(q+1)+1, not divisible by 3 as it leaves a remainder 1

Therefore n is divisible by 3 and n+2 and  n+4 are not divisible by 3.

CASE 2

n= 3q +1 noy divisible by 3 because it leaves a remainder 1

n+2= 3q+1+2, 3q+3, 3(q+1) , divisible by 3 

n + 4 = 3q+1+4 , 3q+5 , 3q +3+2,3(q+1)+2 ,not divisible by 3 because it leavesa remainder 2

Therefore n +2 is divisible by 3 and n and n+4 are not divisible by 3

CASE 3

n=3q+2 ,not divisible by 3 because it leaves a remainder 2

n+2= 3q+2+2 , 3q+4 , 3q + 3+1, 3 (q+1)+1 ,not divisible by 3 as it leaves a remainder 1 

n+4=3q+2+4, 3q+6 ,3(q+2),divisible by 3 

Therefore n+4 is divisible by 3 but n and n +2 are not divisible by 3

Therefor only one out of n , n+2, n+4 is divisible by 3