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

When n = 1, exactly one of 1, 1 + 2, and 1 + 4 is divisible by 3, namely 1 + 2, since 3 is divisible by 3, and the other two 1 and 5 are not.
Suppose for n < k, only one out of n, n+2, n+4 is divisible by 3
For n = k, we consider k, k+2, k+4.  
By the induction hypothesis, only one of k-1, k+1 and k+3 is divisible by 3.  We look at the three possible cases.

Case 1: k-1 is the one which is divisible by 3. Then k-1 = 3m, for some positive integer m.  
Then add 1 to both sides of k-1 = 3m:
k-1+1=3m+1
k = 3m+1
then k is NOT divisible by 3
Now add 3 to both sides of k-1 = 3m:
k-1+3=3m+3
k+2 = 3m+3
k+2 = 3(m+1)
then k+2 is divisible by 3
Now add 5 to both sides of k-1 = 3m:
k-1+5=3m+5
k+4 = 3m+5
k+4 = 3m+3 + 2 = 3(m+1)+2
then k+4 is NOT divisible by 3.
So, we have proved case 1 for n = k

Case 2: k+1 is the one which is divisible by 3. Then k+1 = 3m, for
some positive integer m.  
Then add -1 to both sides of k+1 = 3m:
k+1-1=3m-1
k = 3m-1
then k is NOT divisible by 3
Now add 1 to both sides of k+1 = 3m:
k+1+1=3m+1
k+2 = 3m+1
then k+2 is NOT divisible by 3
Now add 3 to both sides of k+1 = 3m:
k+1+3=3m+3
k+4 = 3m+3
k+4 = 3(m+1)
so k+4 is divisible by 3.
So, we have proved case 2.

Case 3: k+3 is the one which is divisible by 3. Then k+3 = 3m, for some positive integer m.  
Then add -3 to both sides of k+3 = 3m:
k+3-3=3m-3
k = 3(m-1)
then k is divisible by 3
Now add -1 to both sides of k+3 = 3m:
k+3-1=3m-1
k+2 = 3m-1
then k+2 is NOT divisible by 3
Now add 1 to both sides of k+3 = 3m:
k+3+1=3m+1
k+4 = 3m+1
then k+4 is NOT divisible by 3.


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