2. Show that every positive odd integer is of the form (4q+1) or (4q+3), where q is some integer.
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We have
Any positive integer is of the form 4q+1or4q+3
As per Euclid’s Division lemma.
If a and b are two positive integers, then,
a=bq+r
Where 0≤rLet positive integers be a.and b=4
Hence,a=bq+r
Where, (0≤r<4)
R is an integer greater than or equal to 0 and less than 4
Hence, r can be either 0,1,2and3
Now, If r=1
Then, our be equation is becomes
a=bq+r
a=4q+1
This will always be odd integer.
Now, If r=3
Then, our be equation is becomes
a=bq+r
a=4q+3
This will always be odd integer.
Hence proved.
By topper.com
Any positive integer is of the form 4q+1or4q+3
As per Euclid’s Division lemma.
If a and b are two positive integers, then,
a=bq+r
Where 0≤rLet positive integers be a.and b=4
Hence,a=bq+r
Where, (0≤r<4)
R is an integer greater than or equal to 0 and less than 4
Hence, r can be either 0,1,2and3
Now, If r=1
Then, our be equation is becomes
a=bq+r
a=4q+1
This will always be odd integer.
Now, If r=3
Then, our be equation is becomes
a=bq+r
a=4q+3
This will always be odd integer.
Hence proved.
By topper.com
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