Question

Use Euclid's Division lemma show that the every odd tre integer is of the form 4q+1 or 4q+3 for some integer?

Answer 1

AnswEr:

Let us Consider that a & b are two positive integers.

Now, By Using Euclid's Division Lemma

Then, a = bq + r

Here, [ 0 < r = < b]

• b = 4

So, r can be 0, 1, 2 & 3

If r = 0

= a = 4q + 0

= a = 4q

This is an even integer.

If r = 1

= a = 4q + 1

This is an odd Integer.

If r = 2

= a = 4q + 2

This is an even integer.

If r = 3

= a = 4q + 3

This is an odd Integer.

$\rule{150}2$

So, any odd integers is in the form of 4q + 1 & 4q + 3.

Hence, Proved!

Answer 2

Answer:

Step-by-step explanation:

To Prove :-

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

Solution :-

Using Euclid's Division lemma, a = bq + r.

Let a and b be the positive integers.

Then, r = 0, 1, 2 and 3.

If r = 1,

a = bq + r

a = 4q + 1

This will always be odd integer.

If r = 3

a = bq + r

a = 4q + 3

This will always be odd integer.

Hence, Any positive integer is of the form 4q + 1 or 4q + 3.