Show that every positive even integer is of the form 2 q and every positive odd integer is of the form 2 q + 1, where q is a whole number.

Foundation

Mathematics Foundation

Properties of Real Number

Rational Numbers

Playing with Numbers

Question

Show that every positive even integer is of the form

2 q

and every positive odd integer is of the form

2 q + 1

, where q is a whole number.

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(i) Let '

a

' be an even positive integer.
Apply division algorithm with a and b, where

b = 2

a = (2 \times q) + r

where

0\le r &lt; 2

a = 2 q + r

where

r = 0

r = 1

since '

a

' is an even positive integer, 2 divides '

a

∴ r = 0 \Rightarrow a = 2 q + 0 = 2 q

Hence,

a = 2 q

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 \times q) + r

where

0\le r &lt; 2

a = 2 q + r

where

r = 0

or 1
Here

r \neq 0

(

∵ a

is not even)

\Rightarrow r = 1

∴ a = 2 q + 1

Hence,

a = 2 q + 1

when '

a

' is an odd positive integer.