Question

show that every positive even integer is of the form 2q, and that very positive odd integer is of the form 2q+1, where q is some integer.​

Answer 1

Let m be any positive integer which is divided by 2; q and r are the quotient and remainder respectively; then,

By Euclid's division algorithm,

m = 2q + r where -1 < r < 2

=) m = 2q = 2(q) or, m = 2q+1

Clearly, 2q is the even value of m as it is the multiple of 2 and 2q + 1 is the odd value of it as it is the successor of an even number.

Hence, it is proved that every positive even integer is of he form 2q, and that every positive odd integer is of the form 2q +1 for q = {Z}.

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