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Show that every positive even integer is of the form2q and every positive odd integer is of the form
Asked by suryap2 | 26 May, 2010, 04:16: PM Expert Answer

Dear Student,

According to Euclid’s Division Lemma,

Given positive integers a and b, there exist unique integers q and r

satisfying a = bq + r,    0 ≤ r < b

This lemma is true for any two integers

Now, 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. (Note that 'even integer + 1' is always an odd integer)

Hence, 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

Regards,

Team

Topper Learning

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