Question

# Show that one and only one out of n, n+2, n+4 is divisible by 3, where n is any positive integer.

Solution

Let q be the quotient and r be the remainder when n is
divided by 3.
Therefore, n=3q+r, where r=0,1,2
>> n=3q or n=34+1 or 3q +2
Case(i) if n=39, then n is divisible by 3, n+2 and n+4 are
not divisible by 3.
Case (ii) ifn=39+ 1 then n+2=3q+3 =3(q+1), which is
divisible by 3 and n +4 = 3q +5, which is not divisible by 3.
So, only (n +2) is divisible by 3.
Case (iii) if n = 39 + 2, then n + 2 = 39 + 4, which is not
divisible by 3 and (n+4)=39 +6=3(q+2), which is divisible
by 3.
So, only (n + 4) is divisible by 3.
Hence one and only one out of n, (n+2), (n+4) is divisible
by 3.