Question

4.
Show that every positive even integer is of the form 2q and
that every positive odd integer is of the form 2q +1, where q
is some integer​

Answer 1

Answer:

If we take 2 as a positive even integer for example,

Then we get, 2(2)= 4

Next if we take 3 as a positive odd integer for example,

Then we get, 2(3)+1= 7

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Answer 2

Answer:

Let 'a' be an even positive integer.

Apply division algorithm with a and b, where

$b = 2$

$a = (2 \times q) + r$

where 0≤r<2

$a = 2q + r \:$

where r = 0 and r = 1

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

$a = 0 = > 2q$

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

$b = 2$

$a = 2q + r$

where 0≤r<2

$a = 2q + 1 \: \: \: \: \: \: (odd)$

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

Hence, Proved.

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NaVila11

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