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Show that and only one out of n, n+2 , n...

Show that and only one out of n, n+2 , n+4, is divisble by, 3 , where n is any postivie integer.

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On dividing n by 3, let q be the quotient and r be the remainder.
Then , n+ 3q +r, where ` 0 le r gt 3`
` Rightarrow n = 3q +r ," where" r= 0,1,2`
` Rightarrow n = 3q or n = 3q +1 or n = 3q +2`
Case I if n = 3q then n is divisible by3.
Case II If n = 3q +1 then ( n +2) =3q +3 = 3( q +1) , which is divisible by 3.
So , in the case, (n +2) is divisible by 3.
Case III when n = 3q +2 then (n+4) = 3q + 6 =3 ( q+2), which is divisible by 3.
Hence, one and only one out of n , n+2, n+4 is divisible by 3.
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