Question

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

Answer 1

Answer:

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

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

a=(2×n)+r where 0≤r<2

a=2n+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=2n 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×n)+r where 0≤r<2  

a=2n+r where r=0 or 1  

Here r  is not equal to 0 (∵a is not even) ⇒r=1

∴a=2n+1

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