Question

prove that any positive odd integer in either of the form 4q+1 or 4q+3 also show that n2-1 is divisible by 8 it n is odd

Answer 1

Hi!

Here is the answer to your question.

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Let a be any positive integer and b = 4. Then by Euclid’s algorithm, 

a = 4q + r  for some integer q ≥ 0, and r = 0, 1, 2, 3 because 0 ≤ r < 4

So, a = 4q or 4q + 1 or 4q + 2 or 4q + 3

Now, 4q i.e., 2(2q) is an even number

∴4q + 1 is an odd number.

4q + 2 i.e., 2(2q + 1) which is also an even number.

∴ (4q + 2) + 1 = 4q + 3 is an odd number.

 

Thus, we can say that any odd integer can be written in the form 4q + 1 or 4q + 3 where q is some integer.

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 Hope it helps u !!!

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