Euclid's Division Lemma

Euclid’s division lemma states there are two positive integers a and b, where a ≥ b, there exists a unique set of integers q and r, such that:

a = bq + r

This lemma is the foundation of the Euclidean algorithm, used for computing the GCD of two numbers.  
 

The statement of Euclid’s division lemma 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 q is the quotient and r is the remainder, 0 ≤ r < b.

For the proof of Euclid’s division lemma, let us consider the following arithmetic progression as: …, a - 3b, a - 2b, a - b, a, a + b, a + 2b, a + 3b,...

This arithmetic progression has a common difference ‘b’ and extends indefinitely in both directions. Now let us 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 a multiple of the common difference ‘b’ since both are in the arithmetic progression.

So we can write the arithmetic progression as:

a - r = bq

a = bq + r

where ‘r’ is the smallest non-negative integer, therefore, 0 ≤ r < b.

We will now prove the uniqueness of q and r:

Let us consider another pair q’ and r’ such that a = bq’ + r’ and 0 ≤ r’ < b., then we have:

bq + r = bq’ + r’

b(q - q’) = r - r’

Since 0 ≤ r’ < b and 0 ≤ r < b.

Since, |r’ - r| < b and b divides (r - r’)

Therefore, if q = q’ and r = r’

Hence, it is proved that q and r are unique. 

Euclid’s division lemma is used to find the HCF of two given numbers. The steps to find the HCF of two numbers using Euclid’s division lemma is given below:

Let us take two numbers x and y for which we have to find the HCF using Euclid’s division lemma, such that x > y.

Step 1: First, we have to apply the Euclid’s division lemma to ‘x’ and ‘y’. We can find whole numbers, ‘q’ and ‘r’, such that x = yq + r, where 0 ≤ r < y.

Step 2: If r = 0, ‘y’ is the HCF of ‘x’ and ‘y’. If r is not equal to 0, apply the division lemma again to ‘y’ and ‘r’.

Step 3: Till the remainder is 0, this division continues. The divisor at this stage will be the required HCF.

Mastering Euclid’s Division Lemma becomes easier with a few simple strategies. These tips help you apply the lemma quickly and accurately in solving division and HCF-related problems.

 

• Always remember the form a = bq + r, where 0 ≤ r < b.
  
   
• a is the dividend, b in the divisor, q is the quotient, and r is the remainder.
  
   
• Always ensure the remainder is smaller than the divisor.
  
   
• Practice with small numbers first in order to better grasp the pattern before moving to larger ones.
  
   
• If a number divides the product of two numbers, you can use the division lemma to check divisibility and solve problems involving factors and multiples efficiently.

Euclid's division lemma has practical uses in everyday life. Its applications vary from helping in dividing resources to technology enhancements. Let us take a look at some such uses.

 

• Finding HCF for Sharing: Helps in dividing items into equal groups without leftovers, like distributing sweets, pencils, or resources evenly among children or teams.
  
   
• Simplifying Fractions: Used to reduce fractions to their simplest form by finding the HCF of numerator and denominator.
  
   
• Scheduling Repeated Events: Helps determine common intervals for repeating events, like bus timings or rotating work shifts, using HCF.
  
   
• Cryptography and Security: Forms the basis for algorithms in encryption and secure communication, where number theory and HCF calculations are essential.
  
   
• Puzzle and Game Design: Useful in designing games and puzzles involving grouping or arranging objects in patterns without remainder, ensuring fairness and balance.