Euclid's Division Algorithm is a method used to find the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of two positive integers.
- It works by repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder becomes 0.
- The last non-zero remainder is the GCD of the two numbers.
Question 1: Find the GCD of 315 and 84 using the Euclidean Division Algorithm.
Solution:
Given, a = 315 and b = 84
Divide 315 by 84.
315 = 84 × 3 + 63
Replace a with 84 and b with 63.
84 = 63 × 1 + 21
Replace a with 63 and b with 21.
63 = 21 × 3 + 0
Stop because the remainder is 0.
Therefore, GCD(315, 84) = 21
Question 2: Find two integers x and y such that 99x + 78y = GCD(99, 78)
Solution:
First, find the GCD using Euclid's Algorithm.
99 = 78 × 1 + 21 ...(i)
78 = 21 × 3 + 15 ...(ii)
21 = 15 × 1 + 6 ...(iii)
15 = 6 × 2 + 3 ...(iv)
6 = 3 × 2 + 0 ...(v)
So, GCD(99, 78) = 3
Now express 3 as a linear combination of 99 and 78.
From (iv), 3 = 15 − 6 × 2
From (iii), 6 = 21 − 15
Substitute, 3 = 15 − (21 − 15) × 2
3 = 15 × 3 − 21 × 2
From (ii), 15 = 78 − 21 × 3
Substitute, 3 = (78 − 21 × 3) × 3 − 21 × 2
3 = 78 × 3 − 21 × 11
From (i), 21 = 99 − 78
Substitute, 3 = 78 × 3 − (99 − 78) × 11
3 = 78 × 14 − 99 × 11
Therefore, x = -11 and y = 14.
Question 3: Find the GCD of 924 and 396 using the Euclidean Division Algorithm.
Solution:
Given, a = 924 and b = 396
924 = 396 × 2 + 132
396 = 132 × 3 + 0
Stop because the remainder is 0.
Therefore, GCD(924, 396) = 132.
Question 4: Find two integers x and y such that 65x + 40y = GCD(65, 40)
Solution:
First, find the GCD.
65 = 40 × 1 + 25 ...(i)
40 = 25 × 1 + 15 ...(ii)
25 = 15 × 1 + 10 ...(iii)
15 = 10 × 1 + 5 ...(iv)
10 = 5 × 2 + 0 ...(v)
So, GCD(65, 40) = 5
Now express 5 as a combination of 65 and 40.
From (iv), 5 = 15 − 10
From (iii), 10 = 25 − 15
Substitute, 5 = 15 − (25 − 15)
5 = 15 × 2 − 25
From (ii), 15 = 40 − 25
Substitute, 5 = (40 − 25) × 2 − 25
5 = 40 × 2 − 25 × 3
From (i), 25 = 65 − 40
Substitute, 5 = 40 × 2 − (65 − 40) × 3
5 = 40 × 5 − 65 × 3
Therefore, x = -3 and y = 5.
Question 5: Find the GCD of 1029 and 294 using the Euclidean Division Algorithm.
Solution:
Given, a = 1029 and b = 294
1029 = 294 × 3 + 147
294 = 147 × 2 + 0
Stop because the remainder is 0.
Therefore, GCD(1029, 294) = 147.
Practice Questions
Question 1: Find the GCD of 420 and 126, using the Euclidean Division Algorithm.
Question 2: Find two integers x and y such that 56x + 15y = GCD(56, 15), using the Extended Euclidean Algorithm.
Question 3: Prove that GCD(198, 126) = GCD(126, 72), using the Euclidean Division Algorithm.
Question 4: Find the GCD of 735 and 294, using the Euclidean Division Algorithm.
Question 5: Find two integers x and y such that 119x + 34y = GCD(119, 34) ,using the Extended Euclidean Algorithm.