Question

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

Answer 1

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

THANKS

#BeBrainly.

Answer 2

Answer:

Let a be a given positive odd integer.

Applying Euclid's Division Lemma to a and 4.

We get, a = 4q + r, where 0 ≤ r < 4

➡ a = 4q + r, where r = 0,1,2,3

➡ a = 4q or 4q + 1 or 4q + 2 or 4q + 3

But a = 4q and 4q + 2 = 2 (2q + 1) are clearly even.

This, when a is odd, it is of the form

a = 4q + 1 or 4q + 3 for some integer q.

Step-by-step explanation:

@GENIUS