Real numbers- Example 2

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

Solution : Let a be any positive integer and b = 2. Then , by Euclid’s algorithm, a = 2q +r, for some integer q 0, and r =0 or r= 1, because 0r<2. So a= 2q or 2q +1.

If a is of the form 2q, then a ids an even integer . Also, a positive integer can be either even or odd. Therefore, any positive odd integer I sof the form 2q +1.