Question

show that only one out of n,n+1,n+4 is divisible by 3 where n is any positive integer

Answer 1

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 

HOpE It hElpS U ALl !!!...:)

Answer 2

HEY DEAR...
YOUR ANSWER ⬇

Case I : When n = 3q
In this case, we have,
n=3q, which is divisible by 3
n=3q
= adding 2 on both sides
n + 2 = 3q + 2
n + 2 leaves a remainder 2 when divided by 3
Therefore, n + 2 is not divisible by 3
n = 3q
n + 4 = 3q + 4 = 3(q + 1) + 1
n + 4 leaves a remainder 1 when divided by 3
n + 4 is not divisible by 3
Thus, n is divisible by 3 but n + 2 and n + 4 are not divisible by 3
Case II : When n = 3q + 1
In this case, we have
n = 3q +1
n leaves a reaminder 1 when divided by 3
n is not divisible by 3
n = 3q + 1
n + 2 = (3q + 1) + 2 = 3(q + 1)
n + 2 is divisible by 3
n = 3q + 1
n + 4 = 3q + 1 + 4 = 3q + 5 = 3(q + 1) + 2
n + 4 leaves a remainder 2 when divided by 3
n + 4 is not divisible by 3
Thus, n + 2 is divisible by 3 but n and n + 4 are not divisible by 3
Case III : When n = 3q + 2
In this case, we have
n = 3q + 2
n leaves remainder 2 when divided by 3
n is not divisible by 3
n = 3q + 2
n + 2 = 3q + 2 + 2 = 3(q + 1) + 1
n + 2 leaves remainder 1 when divided by 3
n + 2 is not divsible by 3
n = 3q + 2
n + 4 = 3q + 2 + 4 = 3(q + 2)
n + 4 is divisible by 3
Thus, n + 4 is divisible by 3 but n and n + 2 are not divisible by 3 . 
I HOPE YOU FIND IT HELPFUL.