Finding the Highest Common Factor of 506 and 782

The highest common factor (HCF), also known as the greatest common divisor (GCD), of two numbers is the largest positive integer that divides both numbers without leaving a remainder.

To find the HCF of 506 and 782, we can use the Euclidean Algorithm, which is an efficient method for computing the HCF of two integers.

Understanding the Euclidean Algorithm

The Euclidean Algorithm works by repeatedly applying the division lemma:

Given two positive integers, say \(a\) and \(b\), with \(a > b\), we can write \(a = bq + r\), where \(q\) is the quotient and \(r\) is the remainder, such that \(0 \le r < b\). The HCF of \(a\) and \(b\) is the same as the HCF of \(b\) and \(r\). We continue this process until the remainder is 0. The last non-zero remainder is the HCF of the original two numbers.

Applying the Euclidean Algorithm to 506 and 782

Let's find the HCF of 782 and 506 using the steps of the Euclidean Algorithm:

Step 1: Divide 782 (the larger number) by 506 (the smaller number).

\begin{equation*} 782 = 506 \times 1 + 276 \end{equation*}

The remainder is 276.

Step 2: Now, take the previous divisor (506) and the remainder (276). Divide 506 by 276.

\begin{equation*} 506 = 276 \times 1 + 230 \end{equation*}

The remainder is 230.

Step 3: Take the previous divisor (276) and the remainder (230). Divide 276 by 230.

\begin{equation*} 276 = 230 \times 1 + 46 \end{equation*}

The remainder is 46.

Step 4: Take the previous divisor (230) and the remainder (46). Divide 230 by 46.

\begin{equation*} 230 = 46 \times 5 + 0 \end{equation*}

The remainder is 0.

Since the remainder is now 0, the process stops. The HCF is the last non-zero remainder, which is 46.

Result: HCF of 506 and 782

The highest common factor of 506 and 782 is 46.

Revision Table: Summarizing HCF Calculation Steps

The last non-zero remainder is 46, confirming it as the HCF.

Additional Information on Finding HCF

Another common method to find the HCF is by using prime factorization.

Prime Factorization Method for HCF

To find the HCF using prime factorization:

• Find the prime factors of each number.
• List all the common prime factors.
• Multiply the common prime factors, using the lowest power that each factor appears in the original numbers' factorizations.

For instance, to find the HCF of 506 and 782 using this method, you would first find their prime factorizations:

• Prime factorization of 506: \(2 \times 11 \times 23\)
• Prime factorization of 782: \(2 \times 17 \times 23\)

The common prime factors are 2 and 23. The lowest power of 2 is \(2^1\) and the lowest power of 23 is \(23^1\).

HCF = \(2 \times 23 = 46\).

Both the Euclidean Algorithm and the prime factorization method lead to the same HCF. The Euclidean Algorithm is often more efficient for larger numbers as it avoids the need to find prime factors.