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Chapter 6: Problem 7

Find the greatest common factor for each list of terms. \(16 y, 24\)

Short Answer

Expert verified
The greatest common factor is 8.

Step by step solution

01

- Find Prime Factors

First, find the prime factors of each term.For 16y: \[16y = 2^4 \times y\]For 24: \[24 = 2^3 \times 3\]
02

- Determine Common Factors

Next, identify the common prime factors from the factorizations. Both 16y and 24 have prime factors of 2. The smallest power of 2 that appears in both factorizations is \(2^3\).
03

- Calculate Greatest Common Factor

The greatest common factor (GCF) is determined by multiplying the common prime factors raised to their smallest powers.So, the GCF is:\[2^3 = 8\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factors
Prime factors are the building blocks of numbers. They are prime numbers that multiply together to make another number.
For example, the prime factors of 16 are 2 and 2 and 2 and 2, written as \(2^4\).
To find the prime factors of any number, you keep dividing the number by the smallest prime number (starting with 2) until you get a prime number as the result.
For 24, you start by dividing by 2 since it is even:
\24 梅 2 = 12\
Keep dividing:
\12 梅 2 = 6\ and then \6 梅 2 = 3\.
Now you cannot divide further by 2. The final result is 3, which is also a prime number.
So, the prime factors of 24 are \(2 \times 2 \times 2 \times 3\), or written as \(2^3 \times 3\).
Common Factors
Common factors are factors that two or more numbers share. If you find the prime factors of two numbers, common factors are those that appear in both. For instance in the example, we've already found that:
For 16y: \[16y = 2^4 \times y \]
For 24: \[24 = 2^3 \times 3 \]
Here, the common factor is the prime number 2, which appears in both factorizations. It is important to remember that we need the smallest power of the common prime.
In both terms, the smallest power of 2 that appears is \(2^3\). So \(2^3\) is the common factor.
GCF Calculation
Calculating the Greatest Common Factor (GCF) involves multiplying the common prime factors raised to their smallest powers. From our example with the factors of 16y and 24:
The common factor is \(2^3\).
The GCF is then calculated as:
\[2^3 = 8\]
This means the GCF of 16y and 24 is 8.
The GCF is useful because it helps simplify fractions or any forms of equations where common factors are involved. It also plays a key role in solving problems related to divisibility, ratios, and more!

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