Question

Find the HCF of 120 and 156 using Euclid's division algorithm. (1) 18 (2) 12 (3) 6 (4) 24

Short answer

Answer: 12

Step-by-step solution

Understand Euclid's Division Algorithm

Euclid's Division Algorithm states that for any two positive integers 'a' and 'b', the HCF can be found by iteratively applying the division operation using division, remainder, and divisor until the remainder becomes zero. The divisor at this stage will be the HCF of the two numbers.

Apply the First Step of Euclid's Division Algorithm

First, divide the larger number (156) by the smaller number (120), and compute the remainder. 156 = 1 * 120 + 36 Here, the remainder is 36.

Continue Applying Euclid's Division Algorithm

Now, divide the previous divisor (120) by the previous remainder (36) and compute the new remainder. 120 = 3 * 36 + 12 The new remainder is 12.

Keep Applying Euclid's Division Algorithm

Continue the process from Step 3. 36 = 3 * 12 + 0 This time, the remainder is 0.

Identify the HCF

Since the remainder has become 0, the divisor from the last step (12) is the HCF of the two numbers. Hence, the HCF of 120 and 156 is 12. The correct answer is option (2) 12.

Key concepts

HCF (Highest Common Factor)

The Highest Common Factor (HCF), also known as the greatest common divisor, is the largest number that can exactly divide two or more integers without leaving a remainder. HCF plays a critical role in simplifying fractions, finding equivalencies, and solving problems that require identifying common attributes or quantities.

For example, when dealing with fractions, knowing the HCF allows you to reduce them to their simplest form, making calculations easier. The concept of HCF is also essential in areas such as number theory and cryptography.

Division Operation

The division operation is a fundamental arithmetic process that involves dividing a number, known as the dividend, by another number, called the divisor, to obtain a quotient and possibly a remainder. When we perform division on two whole numbers, if the divisor does not divide the dividend completely, a remainder is produced.

In Euclid's Division Algorithm, the division operation is used iteratively to get progressively smaller remainders until we reach a remainder of zero. At that point, the divisor that led to a zero remainder is the HCF of the original two numbers being compared.

Divisor and Remainder

In the context of division, the divisor is the number by which the dividend is divided, while the remainder is what is left over after the division operation. A key aspect of Euclid's Division Algorithm is that the remainder must always be smaller than the divisor, which is why with each step of the algorithm, the remainder becomes the new divisor, ensuring a sequence of decreasing positive integers.

The connection between divisor and remainder is such that when the remainder reaches zero, the last non-zero remainder is the HCF of the original numbers. This relationship is key to finding the HCF efficiently without listing all the factors of the given numbers.

Iterative Algorithms

Iterative algorithms are a set of processes that repeat steps until a certain condition is met. In the case of Euclid's Division Algorithm, the condition is that the remainder should become zero. The beauty of iterative algorithms like this one lies in their systematic, step-by-step approach, which often provides a more computationally efficient and faster solution compared to other methods.

Iterative algorithms are commonly used in computer science and mathematics to solve problems that may otherwise be exceedingly complex or time-consuming. They break down complex problems into smaller parts that are easier to manage and solve incrementally.