Question

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

Answer 1

dear friend

your answer :::-------;-----

b = 4q+1 , or 4q+2
r = 0, 1, 2 ,3

if r = 0

a = 4q + 0
it is not in form of 4q+1

if r = 1

a = 4q+1

it is in the form of 4q+1

if r = 2

a= 4q+2
a = 2( 2q+1 )
it is not in the form of 4q+3

if r = 3

a= 4q+3
it is in the form of 4q+3 .

hope it will help you

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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 .