Question

Show that every positive odd integers is the form(4q+1)or (4q+3) for some integer

Answer 1

heya

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Here is your answer 

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Let a be any positive integer 

B =4 

a = bq+r       ⇒0 ≤ r <b  ⇒ 0≤ r <4 

PUT r = 1 , 2 ,3       ⇒ a = 4q +1 or a = 4q +2 or a = 4 q +  3 

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Hope you satisfied with my answer 

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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 .