Square of an odd integer

Asked by fathima12 | 26th Apr, 2010, 06:09: PM

Expert Answer:

Dear student,

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

6q+0

or

6q+1

or

6q+2

or

6q+3

or

6q+4

or

6q+5

 

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.

 

Regards Topperlearing.

Answered by  | 28th Apr, 2010, 01:04: AM

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