ILLUSTRATIVE EXAMPLESEXAMPLE1 Show that every positive even integer is

1.	ILLUSTRATIVE EXAMPLESEXAMPLE1 Show that every positive even integer is of the form 24, and that every positive odINCERT], 1,-division lemma thereutcrer is of the form 2q + 1, where q is some integer
Answer» Let a be any positive integer andb = 2.Then, by Euclid’s division lemma there exist integers q and r such that A = 2q + r, where 0 ≤ r ≤ 2 Now, o ≤ r ≤ 2 → r = 0 or, r = 1 .˙. 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. Therefore, any odd integer is of the form2q+1.

ILLUSTRATIVE EXAMPLESEXAMPLE1 Show that every positive even integer is of the form 24, and that every positive odINCERT], 1,-division lemma thereutcrer is of the form 2q + 1, where q is some integer

Answer»

Let a be any positive integer andb = 2.Then, by Euclid’s division lemma there exist integers q and r such that

A = 2q + r, where 0 ≤ r ≤ 2

Now, o ≤ r ≤ 2 → r = 0 or, r = 1

.˙. 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. Therefore, any odd integer is of the form2q+1.