Square of an odd integer
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.
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