Prove thatvany positive odd integer is of the form 4q + 1 or 4q + 3

1.	Prove thatvany positive odd integer is of the form 4q + 1 or 4q + 3
Answer» Let\xa0\xa0be any positive integerWe know by Euclid\'s algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying,\xa0where.Take\xa0Since 0 ≤\xa0r\xa0< 4, the possible remainders are 0, 1, 2 and 3.That is,\xa0\xa0can be\xa0, where\xa0q\xa0is the quotient.Since\xa0\xa0is odd,\xa0\xa0cannot be 4q\xa0or 4q\xa0+ 2 as they are both divisible by 2.Therefore, any odd integer is of the form 4q\xa0+ 1 or 4q\xa0+ 3.

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