Question

show that for any postive odd integer is of form 4q+1 or 4q+3,where q is some integer

Answer 1

Let 'a' be any odd positive integer and b=4. By division lemma there exists integers 'q' and 'r' such that
a= 4q + r, where 0 < r <4.
a=4q or a =4q+1 or a =4q +2 or a=4q+3 (therefore 0 < r <4 = r=0, 1, 2, 3)
=a=4q +1 or a=4q +3 (since 'a' is an odd integer therefore a is not equal to 4q, a is not equal to 4q +2)
Hence any odd integer is of the form 4q+1 or 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 .