Question

show that any positive odd integers is of the form 4q+1 or 4q +3 where q is some integers​

Answer 1

let 'a' be the any positive integer

b = 3,

Euclid's division lemma

a = bq+r , 0 less than or equal to 'r' less than 'b'

a = 6q+r , 0 less than or equal to 'r' less than '4'

Therefore, the possible remainders = 0,1,2,3

If r = 0,

a = 4q+0

a = 4q ( even )

If r = 1,

a = 4q+1 ( odd )

If r = 2,

a = 4q+2 ( even )

If r = 3,

a = 4q+3 ( odd )

Therefore, 4q+1, 4q+3 are odd positive integers

Therefore,

Any positive odd integers is of the form 4q+1 or 4q+3 , where 'q' is some integers.