Question

so that euery add intger is in from of 4x+1 and 4x+3​

Answer 1

Correction:

Show that every positive odd integer is of the from 4q+1 and 4q+3.

Solution:

Let 'a' be any positive odd integer and b=4.

Now,

Applying Euclid's Division Lemma,

a=bq+r,0≤r<b

=>a=4q+r,0≤r<4

Implies that r is equal to or greater than zero and lesser than 4.

So,r=0,1,2 and 3

Putting values of r,

Case 1:

When r=0,

a=4q [Even]

Case 2:

When r=1,

a=4q+1 [Odd]

Case 3:

When r=2,

a=4q+2 [Even]

Case 4:

When r=3,

a=4q+3 [Odd]

As,‘a’ is a positive odd integer,

a≠4q

and,a≠4q+2

Thus,every positive odd integer is of the form 4q+1 or 4q+3

Hence,proved