Show that one and only one …

Any positive integer is of the form 5q , 5q + 1 , 5q + 2

here ,
b = 5
r = 0 , 1 , 2 , 3 , 4

when r = 0 , n = 5q
n  = 5q              ----> divisible by 5               ===> [1]
n + 4 = 5q + 4    [ not divisible by 5 ]
n + 8 = 5q + 8 [ not divisible by 5 ]
n + 6 = 5q + 6 [ not divisible by 5 ]
n + 12 = 5q + 12 [ not divisible by 5 ]

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when r = 1 , n = 5q + 1
n = 5q + 1  [ not divisible by 5 ]
n + 4 = 5q + 5 = 5 [q+ 1]    ----> divisible by 5 ===> [2]
n + 8 = 5q + 9 [ not divisible by 5 ]
n + 6 = 5q + 7 [ not divisible by 5 ]
n + 12 = 5q + 13 [ not divisible by 5 ]

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when r =  2 , n = 5q + 2
n = 5q + 2  [ not divisible by 5 ]
n + 4 = 5q +  6 [ not divisible by 5 ]
n + 8 = 5q +10 =  5 [q + 2 ]   --->  divisible by 5   ====> [3]
n + 6 = 5q +8 [ not divisible by 5 ]
n + 12 = 5q + 14 [ not divisible by 5 ]

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when r = 3 , n = 5q + 3
n  = 5q + 3   [ not divisible by 5 ]         
n + 4 = 5q + 7    [ not divisible by 5 ]
n + 8 = 5q + 11 [ not divisible by 5 ]
n + 6 = 5q + 9 [ not divisible by 5 ]
n + 12 = 5q + 15 = 5 [ q + 3 ]  --->  divisible by 5   ====> [4]
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when r = 4 , n = 5q + 4  
n  = 5q + 4   [ not divisible by 5 ]         
n + 4 = 5q + 8    [ not divisible by 5 ]
n + 8 = 5q + 12 [ not divisible by 5 ]
n + 6 = 5q + 10 = 5 [ q + 2 ]  --->  divisible by 5   ====> [5]
n + 12 = 5q + 16 [ not divisible by 5 ]

from 1 , 2 , 3 , 4  , 5 its clear that one and only one out of n, n+4, n+8, n+12 and n+6 is divisible by 5