Question

show that any positive odd integer is form of 4q +1or 4q +3​

Answer 1

Answer:

let us consider a positive odd integer as ‘a’.

On dividing ‘a’ by 4, let ‘q’be the quotient and ‘r’ be the remainder.

Using Euclid's lemma, we have :a=4q+r where 0≤r<4 i.e., r= 0,1,2,3 i.e.,

a= 4q+0 = 4q or a=4q+1

or a=4q+2 or a=4q+3

But, a=4q, a=4q+2 are even values of ‘a’.

But ‘a’ being an odd integer, we have:

a=4q+1, or a=4q+3