Understanding the HCF Problem

The question asks us to find the Highest Common Factor (HCF) of three numbers: 2091, 3485, and 4879. The HCF is the largest positive integer that divides all three numbers without leaving a remainder. Once we find this HCF, let's call it 'x', we need to calculate the sum of its digits.

Calculating HCF Using Euclidean Algorithm

The Euclidean algorithm is an efficient method for computing the HCF of two numbers. To find the HCF of three numbers, we first find the HCF of the first two numbers, and then find the HCF of the result and the third number.

Step 1: Find HCF of 2091 and 3485

We apply the Euclidean algorithm to the numbers 3485 and 2091:

Divide 3485 by 2091:

\( 3485 = 1 \times 2091 + 1394 \)

Now, divide the divisor (2091) by the remainder (1394):

\( 2091 = 1 \times 1394 + 697 \)

Next, divide the divisor (1394) by the remainder (697):

\( 1394 = 2 \times 697 + 0 \)

Since the remainder is 0, the HCF of 2091 and 3485 is the last non-zero remainder, which is 697.

Step 2: Find HCF of 697 and 4879

Now we find the HCF of the result from Step 1 (697) and the third number (4879). We apply the Euclidean algorithm to the numbers 4879 and 697:

Divide 4879 by 697:

\( 4879 = 7 \times 697 + 0 \)

Since the remainder is 0, the HCF of 697 and 4879 is the last non-zero remainder, which is 697.

Therefore, the HCF of 2091, 3485, and 4879 is 697.

Calculating the Sum of Digits of x

The problem states that the HCF is x. We found that \( x = 697 \).

Now we need to find the sum of the digits of x.

The digits of x (697) are 6, 9, and 7.

Sum of digits \( = 6 + 9 + 7 \)

Sum of digits \( = 22 \)

The sum of the digits of x is 22.

Summary Table of HCF Calculation

Revision Table: Key Concepts in Finding HCF

Let's quickly review the key mathematical concepts used in this problem:

• Highest Common Factor (HCF): The largest positive integer that divides two or more integers without leaving a remainder. It is also known as the Greatest Common Divisor (GCD).
• Euclidean Algorithm: An efficient method for computing the HCF of two integers. It is based on repeated division and taking remainders. The last non-zero remainder is the HCF.
• Sum of Digits: The result of adding together each individual digit of a number.

Additional Information on Finding HCF

Besides the Euclidean algorithm, the HCF can also be found using the prime factorization method.

Prime Factorization Method for HCF

• Find the prime factorization of each number.
• Identify the common prime factors.
• Multiply the common prime factors, raised to the lowest power they appear in any of the factorizations.

While prime factorization is useful, the Euclidean algorithm is generally more practical and faster for larger numbers like the ones in this problem, as finding prime factors can be time-consuming.

The concept of HCF is fundamental in number theory and has applications in simplifying fractions, solving certain types of equations, and various other mathematical problems.