Since the remainder is now 0, the last non-zero remainder is the GCF. - Carbonext

Understanding the Context

The GCF, also known as the Greatest Common Divisor, is the largest positive integer that divides two or more numbers without leaving a remainder. In other words, it is the greatest number that is a divisor of all the given numbers.

How the Euclidean Algorithm Works

The Euclidean Algorithm leverages division to systematically reduce the problem of finding the GCF of two numbers. The core idea is straightforward:

1. When dividing two numbers a and b (where a > b), use division to find the remainder r.
   
2. Replace a with b and b with r.
   
3. Repeat the process until the remainder becomes zero.
   
4. The last non-zero remainder is the GCF.

Key Insights

Why the Last Non-Zero Remainder Matters

At each step, the remainders decrease in size. Once the remainder reaches zero, the previous remainder is the largest number that divides evenly into all original numbers. This mathematical integrity ensures accuracy and efficiency.

Example:
Let’s find the GCF of 48 and 18.

• 48 ÷ 18 = 2 remainder 12
  
• 18 ÷ 12 = 1 remainder 6
  
• 12 ÷ 6 = 2 remainder 0

The last non-zero remainder is 6, so GCF(48, 18) = 6.

Final Thoughts

Real-World Applications

Understanding this principle helps in simplifying fractions, solving ratios, optimizing resource distribution, and even in cryptography. Knowing the GCF allows for seamless fraction reduction—turning complex numbers like 48/18 into the simplified 6/3.

In summary, since the remainder is now 0, the last non-zero remainder is the GCF. This simple truth underpins one of the most efficient and reliable algorithms in mathematics. Mastering it builds a strong foundation for tackling more advanced concepts in algebra and number theory.

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