Show that every positive even integer is of the form 2q and that every positive odd integer is of the form 2q$+$1, where q is some integer.

Given :

The given positive integer is q.

To do :

We have to show that every positive even integer is of the form 2q and that every positive odd integer is of the form 2q$+$1.

Solution :

By Euclid's division lemma,

If a and b are two positive integers, then,

$a = b q + r$, where $0 \leq r < b$

Let a be the positive integer, and b $=2$, then,

$a = 2 q + r$, where $0 \leq r < 2$

$r = 0 , 1$.

When $r=0$

$a = 2 q + 0$

$a = 2 q $  

2q is an even positive integer.

When $r=1$

$a = 2 q + 1$

$2q + 1$ is an odd positive integer.

Therefore,  every positive even integer is of the form 2q and that every positive odd integer is of the form 2q$+$1.

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Updated on: 10-Oct-2022

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