Square of an odd integer
Asked by fathima12 | 26th Apr, 2010, 06:09: PM
As square of any odd integer is also an odd integer, therefore if we prove that every odd integer can be expressed in the form of (6q+1) or (6q+3) or (6q+5), then that directly implies that square of an odd integer can also be expressed in similar fashion.
By Euclid's lemma,
for any two given integers, A and B, A can be written in the form
A = Bq + r
Where q is some integer and r is some positive integer such that 0 <= r <= (B-1).
In our case, Let A be any integer and B = 6.
Then any integer can be expressed as
Now, 6q is always even, as it always has 2 as its factor.
When r = even (i.e. 0,2,4)
Then (6q+r) also becomes even.
Similarly when r = odd (i.e. 1,3,5)
then (6q+r) also becomes odd.
=> All odd integers can be expressed in the form 6q+1 or 6q+3 or 6q+5
(and similarly all even integers can be expressed in the form 6q or 6q+2 or 6q+4.)
As We stated earlier that square of an odd integer is also odd
therefore, square of an odd integer can also be expressed as 6q+1 or 6q+3 or 6q+5.
Answered by | 28th Apr, 2010, 01:04: AM
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