Question

Show that any positive integers is in the form of 4k+1 or 4k+3,k €z

Answer 1

any odd integer is of the form 4k + 1 or 4k+ 3.

Step-by-step explanation:

Let be any positive integer

We know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying,

a = bq + r

a = number

q = quotient

r = remainder

r less than b

let's take b = 4

a = 4q + r

Since 0 ≤ r < 4, the possible remainders are 0, 1, 2 and 3.

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

4q = 2× 2q

4q+ 2= 2×(2q+1)

hence these two are even

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

replacing q with k

any odd integer is of the form 4k + 1 or 4k+ 3.