Question

Show that any positive odd integers is of the form 4q+1 or 4q+3, where q is some integers

Answer 1

Let, " a " be any odd positive integer. When " a " is divided by 4, we get " q " as quotient and reminder " r ".

By Eucluids,

a = 4q + r

Now, if r = 0 or 1 or 2 or 3

We get, a = 4q or 4q + 1 or 4q + 2 or 4q + 3.

Since, a = odd

Therefore,

a = 4q +1 or 4q + 3

as,

a = 4q or 4q +2 given an even result.

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 .