Question

show that every positive even integer is in the form of 2q, and that every positive odd integer is of the form 2q+1 where q is some integer​

Answer 1

Step-by-step explanation:

(i) Let 'a' be an even positive integer. 

Apply division algorithm with a and b, where b=2 

a=(2×q)+r where 0≤r<2

 a=2q+r where r=0 or r=1

since 'a' is an even positive integer, 2 divides 'a'. 

∴r=0⇒a=2q+0=2q

Hence, a=2q when 'a' is an even positive integer. 

(ii) Let 'a' be an odd positive integer. 

apply division algorithm with a and b, where b=2

a=(2×q)+r where 0≤r<2 

a=2q+r where r=0 or 1 

Here r =0 (∵a is not even) ⇒r=1

∴a=2q+1

Hence, a=2q+1 when 'a' is an odd positive integer.