Question

Compared with Euclid’s Division lemma $ a = bq + r $ then which number is representing the remainder in $ 17 = 6 \times 2 + 5 $ .

Answer 1

Hint: As we know that Euclid’s division lemma is a proven statement that is used to prove other statements in the branch of mathematics. It states that if there are two positive integers $ a $ and $ b $ then, there exists unique integers $ q $ and $ r $ such that it satisfies the condition $ a = bq + r $ , where $ 0 \leqslant r \leqslant b $ . It is also the general form of Euclid’s Division lemma. This statement tells us about the divisibility of integers. In order to solve this we should show that any positive integer $ a $ can be divided by another positive integer $ b $ in such a way that it leaves a remainder $ r $ .

Complete step-by-step answer:
As per the given question we have $ 17 = 6 \times 2 + 5 $ .
Now the Euclid’s division lemma general form is $ a = bq + r $ ,
On comparing both the statements we get that
  $ a = 17,b = 6,q = 2 $ and $ r = 5 $ .
Here $ a = $ Dividend, $ b = $ Divisor, $ q = $ Quotient and $ r = $ Remainder.
So $ r = $ remainder $ = 5 $ .
Hence from the expression $ 17 = 6 \times 2 + 5 $ , remainder is $ 5 $ .
So, the correct answer is “5”.

Note: We should know that the value of remainder i.e. $ r $ should be less than $ q $ i.e. quotient because if remainder is more than quotient than the dividend i.e. $ a $ will not be fully divisible. Remainder can be equal to zero or greater than zero. We should know that Euclid’s Division lemma is only valid for the set of whole numbers and not for the negative integers, fractions irrational numbers or composite numbers. This is because the basic set of numbers that is generally used is a set of whole numbers and Euclid worked only on the set of whole numbers.