Show that every positive integer is of the form 2q , and that every positive odd integer is of the form 2q+1 , where q is some integer.
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To Show :
Every positive odd integer is of the form 2q and that every positive odd integer is of the form 2q+1, where q € Z .
Solution :
Let a be any positive integer.
And let b = 2
So by Euclid's Division lemma there exist integers q and r such that ,
a = bq+r
a = 2q+r (b = 2)
And now ,
As we know that according to Euclid's Division Lemma :
0 ≤ r < b
Here ,
0 ≤ r < 2
Here the possible values of r are = 0,1
=> 0 ≤ r < 1
=> r = 0 or r = 1
a = 2q+0 = 2q or a = 2q+1
And if a = 2q , then a is an integer.
We know that an integer can be either odd or even.
So , therefore any odd integer is of the form 2q+1.
#Hence Proved
Every positive odd integer is of the form 2q and that every positive odd integer is of the form 2q+1, where q € Z .
Solution :
Let a be any positive integer.
And let b = 2
So by Euclid's Division lemma there exist integers q and r such that ,
a = bq+r
a = 2q+r (b = 2)
And now ,
As we know that according to Euclid's Division Lemma :
0 ≤ r < b
Here ,
0 ≤ r < 2
Here the possible values of r are = 0,1
=> 0 ≤ r < 1
=> r = 0 or r = 1
a = 2q+0 = 2q or a = 2q+1
And if a = 2q , then a is an integer.
We know that an integer can be either odd or even.
So , therefore any odd integer is of the form 2q+1.
#Hence Proved
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