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SHOW THAT EVERY POSITIVE EVEN INTEGER IS OF THE FORM 2m AND THAT EVERY POSITIVE ODD INTEGER IS OF THE FORM 2m+1 WHEN m IS SOME INTEGER

 Let a be any positive integer and b = 2

Then by Euclid 's division Lemma, there exist integers q and r such that –

a = 2q + r where 0 ? r < 2

Now 0 ? r < 2 ? 0 ? r ? 1

? r = 0 or r = 1 [? r is an integer]

? a = 2q or a = 2q + 1

If a = 2q, then a is an even integer.

We know that an integer can be either even or odd.So, any odd integer is of the form 2q + 1.


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