Step 1: Problem Definition:
The objective is to determine the Highest Common Factor (HCF) for the numbers 132 and 77.
The HCF is defined as the largest integer that divides both numbers without leaving any remainder.
Step 2: Method Selection:
The Euclidean algorithm is an efficient procedure for calculating the HCF of two integers. It is based on the principle:
\[
\text{HCF}(a, b) = \text{HCF}(b, a \mod b)
\]
This iterative process continues until the remainder of the division is zero. The last non-zero remainder, which is the divisor at that point, is the HCF.
Step 3: Algorithm Application:
Applying the Euclidean algorithm to 132 and 77:
1. Divide 132 by 77:
\[
132 = 1 \times 77 + 55
\]
The remainder is 55. The next step involves finding HCF(77, 55).
2. Divide 77 by 55:
\[
77 = 1 \times 55 + 22
\]
The remainder is 22. The next step involves finding HCF(55, 22).
3. Divide 55 by 22:
\[
55 = 2 \times 22 + 11
\]
The remainder is 11. The next step involves finding HCF(22, 11).
4. Divide 22 by 11:
\[
22 = 2 \times 11 + 0
\]
The remainder is 0. Therefore, the HCF is the divisor at this step, which is 11.
Step 4: Result:
The Highest Common Factor of 132 and 77 is 11.