EUCLID’S DIVISION LEMMA

We have already explored the world of real numbers and encountered irrational numbers. We will continue our discussion on real numbers and learn one of the important properties called Euclid’s division lemma.

LEMMA

A statement that is utilized to prove another statement can be defined as a Lemma.

ALGORITHM

A system of steps that are used to solve a problem is defined as an algorithm. Euclid’s division lemma and algorithm are closely interlinked with each other. The Euclid division lemma/algorithm has several applications related to finding properties of numbers.

EUCLID’S DIVISION LEMMA

STATEMENT: Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

PROOF:

Now, let us try finding the integers q and r for the following pair of positive integers a and b.

(1) 10, 3 (2) 20, 4 (3) 40, 19

Let us write the relation for each pair.

1. Let a = 10 and b = 3

Therefore, 10 = 3 × 3 + 1

Here, q = 3 and r = 1, where 0 ≤ 1 < 3

2. Let a = 20 and b = 4

Therefore, 20 = 4 × 5 + 0

Here, q = 5 and r = 0, where 0 ≤ 0 < 4

3. Let a = 40 and b = 19

Therefore, 40 = 19 × 2 + 2

Here, q = 2 and r = 2, where 0 ≤ 2 < 19

If we observe from the above, for each pair of positive integers a and b, we have found whole numbers q and r, satisfying the relation:

a = bq + r, 0 ≤ r < b

It can also be noticed that q and r are unique. Note that q and r can also be zero.

Hence, it is proved.

Finding HCF By Euclid’s Division Lemma

As we learned above, Euclid's division lemma has several applications related to the properties of numbers. We use Euclid's division lemma to find HCF. We can find the HCF of large numbers, which usually takes a lot of time with general calculations. Euclid's division lemma makes our workflow easier in finding the HCF of a number.

To obtain HCF of two positive integers, say s and t, with s > t, follow the steps below:

Apply Euclid's division lemma to s and t such that there exist unique integers q and r satisfying s = tq + r, 0 ≤ r < t.
If r = 0, then t is the HCF of s and t.
If r ≠ 0, then apply division lemma to t and r.
Continue this process until remainder = 0, and the divisor at this particular stage will be the HCF.

EXAMPLE: Use Euclid’s division lemma to find the HCF of 3814 and 2562.

SOLUTION: Here a = 3814 and b = 2562

Euclid’s division lemma, a = bq + r, 0 ≤ r < b, we get,

3814 = 2562 × 1 + 1252

Since r ≠ 0, continue the process by taking

a = 2562 and b = 1252.

2562 = 1252 × 2 + 58 ( r ≠ 0 )

1252 = 58 × 21 + 34 (r ≠ 0)

58 = 34 × 1 + 24 (r ≠ 0)

34 = 24 × 1+10 (r ≠ 0)

24 = 10 × 2 +4 (r ≠ 0)

10 = 4 × 2 + 2 (r ≠ 0)