Question

Show that any positive odd integer is often form 4q+1, 4q+3 and every even integer is of the form 4q, 4q+2. where q is integer.

Answer 1

hope this help..........

Answer 2

By Euclid’s division algorithm,

a = bq + r

Take b = 4

Since 0 ≤ r < 4, r = 0, 1, 2, 3  

So, a = 4q, 4q + 1, 4q + 2, 4q + 3

Clearly, a = 4q, 4q + 2 are even, as they are  divisible by 2. Therefore 'a' cannot be 4q, 4q +  2 as a is odd. But 4q + 1, 4q + 3 are odd, as  they are not divisible by 2.

Any positive odd integer is of the form  (4q + 1) or (4q + 3)