Question

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

Answer 1

Answer:

Show that the square of any positive odd integer is of the form 4m+1,for same integer n. Let 'a' be any positive integer. According to the question, when b = 4. Therefore, the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.

hope it helps you

Answer 2

Step-by-step explanation:

We have

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.