To find the Highest Common Factor (HCF) of 168 and 126, we'll utilize the Euclidean algorithm, which is a method for finding the HCF of two numbers. Here’s a step-by-step breakdown of the process:
1. Step 1: Understand the Euclidean Algorithm
The Euclidean algorithm involves repeated division of the two numbers, where the larger number is divided by the smaller number, and then the remainder from this division replaces the larger number. This process is repeated until the remainder is zero. The last non-zero remainder is the HCF of the two original numbers.
2. Step 2: Start the Division
We begin with the numbers 168 and 126:
- Divide 168 by 126.
- Calculate the quotient and the remainder.
3. Step 3: Calculate the First Remainder
- 168 divided by 126 is 1 with a remainder of 42.
- Now, replace the larger number (168) with 126, and the smaller number (126) with the remainder (42).
4. Step 4: Repeat the Division Process
- Divide 126 by 42.
- Calculate the quotient and the remainder.
5. Step 5: Calculate the Second Remainder
- 126 divided by 42 is 3 with a remainder of 0.
- Since the remainder is now 0, the process stops here.
6. Step 6: Determine the HCF
- The last non-zero remainder obtained in the process is 42.
- Therefore, the HCF of 168 and 126 is 42.
By following these steps accurately, we determine that the Highest Common Factor of 168 and 126 is [tex]\( \boxed{42} \)[/tex].
