Question

show that any positive even integer can be written in the form 6q, 6q+ 2 or 6q+ 4 where q is some integer​

Answer 1

Let m be any positive integer which is divided by 6, q and r are the quotient and remainder respectively.

Then, by Euclid's division algorithm,

m = 6q + r where -1< r < 6

=) m = 6q or, m = 6q + 1 or, m = 6q +2 or, m = 6q + 3 or, m = 6q + 4 or, m = 6q + 5

But, 6q + 1; 6q + 3 and 6q +5 are the odd values of m.

Hence, it is proved that all even positive integers are of the form 6q, 6q+2 and 6q+4 for q being any integer.

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