Question

show that any positive odd enteger is of the form 4q+1 or 4q+3, where q is some integer is​

Answer 1

Answer:

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).

Answer 2

Answer:

Any positive integer is of the form 4q+1or4q+3

As per Euclid’s Division lemma.

If a and b are two positive integers, then,

a=bq+r

Where 0≤r<b.

Let positive integers be a.and b=4

Hence,a=bq+r

Where, (0≤r<4)

R is an integer greater than or equal to 0 and less than 4

Hence, r can be either 0,1,2and3

Now, If r=1

Then, our be equation is becomes

a=bq+r

a=4q+1

This will always be odd integer.

Now, If r=3

Then, our be equation is becomes

a=bq+r

a=4q+3

This will always be odd integer.

Hence proved.