Question

using eculids division lemma sgow that any positive odd intiger is of the form 4q+1 or 4q+3 where q is same integer.

Answer 1

Hey Mate Here Is Your Answer


As per euclids division lemma

If a and b are 2 positive integer, then

a=bq+r
Let positive integer be a and b=4

Hence, a =4q+r
r is an integer greater than or equal to 0 and less than 4 hence r can be either 0,1,2,or 3.

If r=1

Our equation becomes
a=4q+r
a=4q+1
This will always be an odd integer.

If r=3
Our equation becomes
a=4q+r
a=4q+3

This will always be an odd integer.

Therefore, any odd integer is of the form 4q+1 or 4q+3....


Hence proved......

Answer 2

Step-by-step explanation:

Let a be the positive integer.

And, b = 4 .

Then by Euclid's division lemma,

We can write a = 4q + r ,for some integer q and 0 ≤ r < 4 .

°•° Then, possible values of r is 0, 1, 2, 3 .

Taking r = 0 .

a = 4q .

Taking r = 1 .

a = 4q + 1 .

Taking r = 2

a = 4q + 2 .

Taking r = 3 .

a = 4q + 3 .

But a is an odd positive integer, so a can't be 4q , or 4q + 2 [ As these are even ] .

•°• a can be of the form 4q + 1 or 4q + 3 for some integer q .

Hence , it is solved .