Question

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

Answer 1

let's take an example, of even integers :

2= 2(1)

4= 2(2)

26 = 2(13)

28= 2(14) etc

for odd integers :

1 = 2(0)+1

3= 2(1)+1

9= 2(4)+1

25= 2(12)+1 , etc

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Answer 2

Answer:

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

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=0 (∵a is not even) ⇒r=1

∴a=2q+1

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