Question

show that every positive Integer
is of the form 2q+1 where q is
some integer
​

Answer 1

Answer:

Step-by-step explanation:

we will prove in 2 steps

first by defining even integer than by defining odd ones

(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.

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