Problem 26 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. \(14,44,\) and 616

Step-by-Step Solution

Answer

The greatest common factor is 2.

Step 1: Find the Prime Factorization

First, we need to find the prime factorization of each number. - **14**: The prime factorization of 14 is \(2 \times 7\).- **44**: The prime factorization of 44 is \(2^2 \times 11\).- **616**: The prime factorization of 616 is \(2^3 \times 7 \times 11\).

Step 2: Identify Common Prime Factors

Look at the list of prime factors for the three numbers: - **14**: \(2 \times 7\)- **44**: \(2^2 \times 11\)- **616**: \(2^3 \times 7 \times 11\)The common prime factor across all three numbers is \(2\).

Step 3: Determine the Lowest Power of the Common Factor

The lowest power of the common factor \(2\) is \(2^1\). Therefore, the greatest common factor is \(2\).

Key Concepts

Prime Factorization

When we talk about prime factorization, we're referring to breaking down a number into the set of prime numbers that multiply together to form the original number. Think of it like finding the basic building blocks of a number. Prime numbers are those which can only be evenly divided by 1 and themselves, like 2, 3, 5, 7, etc.

For example:

• 14 can be broken down into the prime factors 2 and 7, since 14 equals 2 times 7.
• 44 is composed of 2, 2, and 11, so its prime factorization is 2 squared (2²) times 11.
• 616 breaks down to 2 cubed (2³), 7, and 11.

By expressing each number in terms of its prime factors, it becomes much easier to spot any shared factors amongst a group of numbers.

Common Prime Factors

Once we have the prime factorizations, the next step is identifying which prime numbers are present in all the factorizations. These are the common prime factors. Finding these shared factors is crucial for determining the greatest common factor (GCF).

Taking our earlier example, here's how we identify common factors:

• 14: prime factors are 2, 7.
• 44: prime factors are 2², 11.
• 616: prime factors are 2³, 7, 11.

If we look closely, the number 2 appears as a factor in all three sets of prime factors. Thus, the prime number 2 is our common factor across these numbers.

Lowest Power of Common Factor

After identifying the common prime factors, it's essential to determine the lowest power of these factors for finding the GCF. The reason we seek the lowest power is to ensure that the common number can divide each of the original numbers without leaving a remainder.

Let’s break it down with our example:

• For 14, the power of 2 is 2¹.
• For 44, the power of 2 is 2².
• For 616, the power of 2 is 2³.

The lowest power among these is 2¹ or just 2. This is the greatest common factor because it’s the largest factor that appears in all the numbers’ prime factorizations at the same minimal level.