Question

H.w. show that any positive positive odd integer
is of the form 4q+ 1 or 4q+3 where a is some
integer​

Answer 1

Answer:

Let a be any odd positive integer and b=4, By Euclid's Division Lemma there exists integers q and r such that

a = 4q+r, where 0 $\leq$ r < 4

⇒ a = 4q or, a = 4q+1 or, a = 4q+2 or, a = 4q+3 [∵0$\leq$r<4⇒r = 0,1,2,3]

⇒ a = 4q+1 or, a = 4q+3 [∵a is an odd integer ∴ a ≠ 4q, a ≠ 4q+2]

Hence any odd integer is of the form 4q+1 or, 4q+3