Question

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

Answer 1

any integer can be written as
1.4q,
2.4q+1,
3.4q+2,
4.4q+3,
5.4(q+1) {=4n, n€z},
check which no is/are odd

only 2 and 4 are not divisible by 2.therefore odd.

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 .