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: 

We will use Euclid’s algorithm to prove this.

a = bq + r where 0 ≤ r < b

Let us assume ‘a’ be any positive integer and b = 2.

a = 2q + r, for an integer q ≥ 0

The value of r can be either 0 or 1 [ ∵ 0 ≤ r < 2].

∴ When r is 0, a = 2q + 0 for an even integer as all it is twice of an integer.

or when r is 1, a = 2q + 1 for an odd integer as 1 is added to an even integer

Hence Proved

☛ Check: NCERT Solutions for Class 10 Maths Chapter 1

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

Summary:

Hence shown 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

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