Euclid Division Lemma

Euclid's Division Lemma (or Algorithm) gives the relation between the various components of division. The statement of Euclid's Division Lemma is as follows:

For any two positive integers a and b, where a ≥ b there exists a unique set of integers q and r, such that: 

a = b q + r

Where, 

• q is called the quotient, and 
• r is called the remainder, 0 ≤ r < b.

It is based on the fact that "Dividend = (Divisor × Quotient) + Remainder"

• Euclid's Division Lemma has many Applications related to the Divisibility of Integers
• It can be used to find the HCF of two numbers.

Euclid’s Division Lemma Example

Here, the given numbers are, 39(=a) and 6(=b) we can write it in a = bq + r form as, 39 = 6×6 + 3

where, quotient(q) is 6 and remainder(r) is 3.

Proof Of Euclid Division Lemma

Consider the following arithmetic progression.

………, a − 3b, a − 2b, a − b, a, a + b, a + 2b, a + 3b, ……

The above arithmetic progression has a common difference ‘b′, and it extends indefinitely in both directions.
Now, Let's consider the smallest non-negative term of this arithmetic progression to be r.

The difference between the smallest non- negative term r and a will be in multiple of the common difference 'b' as they both are in A.P.

So we can write it as,

a – r = bq
a = bq + r

Where, r is the smallest non-negative integer satisfying the above result. 
Therefore, 0 ≤ r < b

Thus, we have a = bq + r, where 0 ≤ r < b

Now, to prove the Uniqueness of q and r:

Let's Consider another pair q′ and r′ such that a = bq′ + r′ and 0 ≤ r′<b, then we would have:

bq + r = bq′ + r′
b(q − q′) = r′ − r

Since 0≤r′<b and 0≤r<b,
then, ∣r′−r∣<b

Therefore, the only possible way for this equation to hold true is if q=q′ and r=r′.

Therefore it is proved that q and r are unique .

Thus, for every two numbers a and b we have unique value of q and r such that 'a = bq + r', as defined in the Euclid's Division Lemma.

Solved Examples on Euclid's Division Lemma

Example 1: Find the quotient and remainder when 315 is divided by 17 using Euclid's Division Algorithm.

Solution:

Given: Dividend = 315, Divisor = 17

Using Euclid's Division Lemma, Divide 315 by 17
⇒ 315 = 17 × 18 + 9

Thus, quotient is 18 and remainder is 9.

Example 2: Find the quotient and remainder when 73 is divided by 9 using Euclid's Division Algorithm.

Solution:

Given: Dividend = 73, Divisor = 9

Using Euclid's Division Lemma, Divide 73 by 9
⇒ 73 = 9 × 8 + 1

Therefore, when 73 is divided by 9, the quotient is 8 and the remainder is 1.

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