Question

show that the square of an odd integer is of the form 4q+1 for the some integer q​

Answer 1

Answer:

According to Euclid's Division Lemma

a = bq+r (0≤r<b)

here b = 4 then r = 0,1,2,3

∴ a = 4q = 2m where m = 2q

⇒It is an even no.

a = 4q+1 = 2m +1  where m = 2q

⇒It is an odd no.

∴ a = 4q+2 = 2m where m = 2q + 1

⇒It is a even no.

a = 4q+3 = 2m + 1 where m = 2q + 1

⇒As it can not be expressed in the form of 2m ,it is not a even no.

∴ An odd no. can be expressed as 4q +1 or 4q+3