Question

Can q be negative on Euclid's division lemma??​

Answer 1

Though Euclid division lemma has been stated for all positive integers only But it can be extended for negative integers too.. only 'b' not= 0

Like for both negative integers a & b , or one negative one positive a & b, there exist many integers q & r ( not unique) satisfying a= bq+r, where remainder r is either 0 or smaller than b or even greater than b.

eg: a = b x q + r

& -37 = -3 x 12 -1 , here q= 12, r = -1

Or, -37 = -3 x -12 - 73 , here q= -12 , r = -73

And Since Euclid division lemma has several applications related to finding properties of basic set of numbers. And we know basic set is Natural numbers, which are always positive.

And also Euclid division lemma is used to compute HCF of numbers. And HCF are calculated for positive numbers not negatives..

Above are the reasons, why we consider positive integers

I hope my answer is helpful for you