Question

In Euclid's Division Lemma, when a = bq + r where a, b are positive integers then what values r can take?

Answer 1

Hint – In order to solve this problem you need to know Euclid's Division Lemma. Here we need to know that r should be less that q because if r is more than q the dividend a is not fully divided.

Complete step by step answer:

Euclid’s Division Lemma:
It tells us about the divisibility of integers. It states that any positive integer ‘a’ can be divided by any other positive integer ‘ b’ in such a way that it leaves a remainder ‘r’.

Euclid's division Lemma states that for any two positive integers ‘a’ and ‘b’ there exist two unique whole numbers ‘q’ and ‘r’ such that, a = bq + r, where 0≤r
Here, a= Dividend, b= Divisor, q= quotient and r = Remainder.

Hence, the values 'r’ can take 0≤r
Note – For example when the number 17 is divided by 3 it can be written as 17 = 3(5)+2. Here the 17 is fully divided therefore we can see 2 (remainder) is less than 5 (quotient).

Euclid's Division Lemma is only valid for the set of whole numbers and not for negative integers, fractions, irrational numbers and composite numbers. This is probably because the basic set of numbers that we deal with are whole numbers and Euclid worked only on them for the same purpose.