Question

show that every +ve even integer is of 2q and every +ve odd integer in form of 2q + 1, where q is some integer.

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Answer 1

In case of 4 , it is a +ve even integer

So , if we do 2q

then it would be 2*4=8

and it is positive and even so we can write it as 2q

but as 2q+1 we write it will be,

2*4+1=9

and ot is not even

so it is clear that even +ve integers can be wrote as 2q and odd +ve integers as 2q+1

Answer 2

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$\leadsto$Let a be any positive integer and b = 2.

$\leadsto$Then, by Euclid's algorithm a = 2q + r. for some integer q >or =, and r = 0 or r= 1, because 0< or = ,r < or =2..

$\leadsto$So, a= 2q or 2q + 1.

$\leadsto$If a is of the form 2q, then a is an even integer. Also, a positive integer can be either even or odd.

$\leadsto$Therefore. any positive odd integer is of the form 2q+ 1

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