Question

Show that ny possitive odd integer is of the formof 4q+1 or 4q+3 where q is the sum interger

Answer 1

to understand this problem...u shud hv knowledge of Euclid division lemma and general form of an odd number which is 2m + 1 i .e Every Positive odd interger Can be expressed of the form 2m + 1 where m is any positive integer .

Answer 2

Let a be any positive integer and b = 4

By Euclid's division lemma - If a and b are two positive integers there exist a unique integers q and r which satisfy a = bq + r , where 0  ≤ r  < b .

∵  a = bq + r
⇒ a = 4q + r
Since, 0  ≤ r  < b ,the possible the possible remainders are 0, 1, 2, 3.

Now we can write a = 4q + 1 or a = 4q + 3
where q is some integer.

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