Euclid Division Lemma

Euclid's Division Lemma states that for any two positive integers a and b, where a ≥ b, there exists a unique set of integers q and r such that: 

a = bq + r

Where

• a = Dividend
• b = Divisor
• q = Quotient
• r = Remainder
• 0 ≤ r < b

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

For example, consider the division of 39 by 6, as shown in the image. We can express it in the form a = bq + r as 39 = 6 × 6 + 3

Similarly, the division lemma can be applied to other numbers, as shown below.

Proof

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

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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