Question

show that any positive even integer is of the form 2q and positive odd integer of the form 2q+1 where q is some integer.

Answer 1

hey!!
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we Let a be any positive integer and b=2.

Then, by Euclid's division lemma there exist integers q and r such that:-

a = 2q + r , where 0 </= r < 2

▪now, 0 </= r <2

=> 0</= r</= 1

=> r=0 or, r=1 ( because r is an integer)

▪therefore a =2 q, then a is an even integer.

▪we know that an integer can either be even or odd. Therefore any odd integer is of the form 2q +1.
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hope help u

Answer 2

$\huge \bf{ \red{hey}}$

$\huge{ \mathfrak{here \: is \: answer}}$

let a be any positive integer

then

b=2

0≤r<b

0≤r<2

r=0,1

case 1.

r=0

a=bq+r

2q+0

2q

case 2.
r=1
a=bq+r

2q+1

from above it is proved.




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