Using Euclid’s divison lemma -
a = bq + r;
We want to show that any even integer is of the form '2q' and any odd integer is of the form'2q + 1'
So, we take the two integers a and 2.
When we divide a with 2, the possible values of remainder are 0 and 1, which means
a = 2q + 0 or a = 2q + 1
Also, 2q is always an even integer for any integral value of q and so 2q + 1 will always be an odd integer. (∵ 'even integer + 1' is always an odd integer)
∴ when a is a positive even integer it will always be of the form 2q
And when a is positive odd integer it will always be of the form 2q + 1.
∴ for every positive integer it is either odd or even.
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