prove that one and only one out of n, n+2orn+4 is divisible by 3 where n is any positive integer

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Solution

$We know that any positive integer is of the form of 3 q, 3 q + 1 or 3 q + 2 for some integer q . CASE 1 : When n = 3 q Now, n = 3 q, that is divisible by 3 . n + 2 = 3 q + 2, that is not divisible by 3 as it leaves a remainder of 2 . n + 4 = 3 q + 4 = 3 (q + 1) + 1, that is not divisible by 3 as it leaves a remainder of 1 So, n is divisible by 3, but n + 2 and n + 4 are not divisible by 3 . CASE 2 : When n = 3 q + 1 Now, n = 3 q + 1, that is not divisible by 3 as it leaves a remainder of 1 . n + 2 = 3 q + 3 = 3 (q + 1), that is divisible by 3 n + 4 = 3 q + 5 = 3 (q + 1) + 2, that is not divisible by 3 as it leaves a remainder of 2 So, n + 2 is divisible by 3, but n and n + 4 are not divisible by 3 . CASE 3 : When n = 3 q + 2 Now, n = 3 q + 2, that is not divisible by 3 as it leaves a remainder of 2 n + 2 = 3 q + 4 = 3 (q + 1) + 1, that is not divisible by 3 as it leaves a remainder of 1 . n + 4 = 3 q + 6 = 3 (q + 2), that is divisible by 3 . So, n + 4 is divisible by 3, but n and n + 2 are not divisible by 3 .$

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