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

1
by krabhirami

2015-05-23T18:06:26+05:30

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n,  n + 2,  n + 4

All numbers can be classified in to three groups:
n =  3 k  multiples of 3
= 3 k + 1    ie., when divided by 3,  they give a remainder 1
= 3 k + 2  ie., when divided by 3, give a remainder 2.

If  n = 3 k ,  then  it is divisible by 3.
then  n + 2 = 3 k +2  or n+4 = 3k + 4 is not divisible by 3.

If n = 3 k + 1,  then  n+2= 3k +3  divisible by 3.
then n+4 = 3 k + 5  is not divisible by 5.

If n = 3 k + 2 ,  then  n + 4 = 3 k + 6  is divisible by 3.
then n + 2 = 3 k  + 4 is not divisible by 3.

so exactly one of them only is divisible by 3 for any n.