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Question

Show that any positive even integer can be written in the form 6q,6q+2 and 6q+4 where q is an integer

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Solution

Let a be any positive odd integer and b = 6.
Therefore,
Now, by placing r = 0, we get, a = 6q + 0 = 6q
By placing r = 1, we get, a = 6q +1
By placing, r = 2, we get, a = 6q + 2
By placing, r = 3, we get, a = 6q + 3
By placing, r = 4, we get, a = 6q + 4
By placing, r = 5, we get, a = 6q +5
Thus, a = 6q or, 6q +1 or, 6q + 2 or, 6q + 3 or, 6q + 4 or, 6q +5
But here we assumed that a is even therefore 6q, 6q + 2, 6q +4 are the even integers
Proved

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