Question

show that every positive odd integer is of the form 4q + 1 or 4 q + 3 for some integers q​

Answer 1

Answer:

HOPE THIS BRING A SMILE IN YOUR FACE

Step-by-step explanation:

Let the odd positive integer be 'a'.

To prove every odd positive integer is of the form 4q+1 or 4q+3.

a=bq+r, b>r>or equal to 0

a=4q+r

r=0,1,2,3.

If r=0

a=4q+0

a=4q

a=2(2q)

a=2m, where m is some integer.

Since a is a multiple of 2, it can't be odd.

If r=1

a=4q+1

a=2(2q)+1

a=2m+1

Therefore a is odd.

If r=2

a=4q+2

a=2(2q+1)

a=2m

Since a is a multiple if 2, it can't be odd.

If r=3,

a=4q+3

a=4q+2+1

a=2(2q+1)+1

a=2m+1

Therefore a is odd.