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

Last updated date: 17th Mar 2023
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Hint: According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r ≤ b.

Complete answer: 

As we know by Euclid's Division Lemma,

a = bq + r 

where $0 \leqslant r < b$

Let positive integer be a

And b = 2 

Hence a = 2q + r 

where $0 \leqslant r < 2$

So, either r = 0 or r = 1

So, a = 2q or a = 2q + 1 

If a is of the form 2q, then a is an even integer. Also, a positive be either even or odd. 

Therefore, any positive odd integer is of form 2q + 1.

NOTE: Euclidean division can also be extended to negative dividend (or negative divisor) using the same formula. You can work out a few examples on the same.