Question

show that any positive integer is of form 4q+1 or 4q+3 where q is a positive integer

Answer 1

I am little confused if this question is correctly framed or not.
Even though we can express each no by above value of 4q+1 or 4q+3 which when divided by 4 gives non zero remainder.
but I am not sure how we can express the numbers which when divided by 4 gives 0 remainder.

999 can be expressed as - 4 * 249 + 3
q = 249
but what about-
40 - we cannot have a non zero remainder for this.

Help me to understand the question well.

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 .