Question

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

Answer 1

let the positive odd integer be a

and b=4

a=bq+r where r is less than b/euclid division lemma/

b=4

therefore r=0,1,2,3,

therefore,a=4q+0

a=4q

a=4q+1

a=4q+2

a=4q+3

odd integer

a=4q+1

a=4q+3

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 .