If d is the HCF of 40 and 65, find the value of the integers x and y which satisfy d = 40x + 65y.

HCF (Highest Common Factor) of two numbers is the largest possible number which divides the two numbers exactly without any remainder.

Answer: If d is the HCF of 40 and 65, the values of the integers x and y which satisfy d = 40x + 65y are x = 5, y = -3

We will explain how to find the HCF of 40 and 65

Explanation:

HCF of 40 and 65 by Prime Factorization 

Represent 40 and 65 as a product of its prime factors.

Prime factorization of 40 is 2 × 2 × 2 × 5

Prime factorization of 65 is 5 × 13

The common factor in the prime factorization of 40 and 65 is 5

HCF is the product of the factors that are common to each of the given numbers.

Since the common factor is 5, the HCF of 40 and 65 is 5

Now, we will express 5 as the linear combination of 40 and 65.

5 = 15 - 10 × 1

5 = 15 - (25 - 15) × 1

5 = 15 - 25 × 1 + 15 × 1

5 = 15 × 2 - 25 × 1

5 = (40 - 25 × 1) × 2 - 25 × 1

5 = 40 × 2 - 25 × 2 + 25 × 1

5 = 40 × 2 - 25 × 3

5 = 40 × 2 - (65 - 40 × 1) × 3

5 = 40 × 2 - 65 × 3 + 40 × 3

5 = 40 × 5 + 65 × (-3)

So, 5 = 40(5) + 65(-3)