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Chapter 10: Problem 15

Determine the greatest common factor. 75 and 25

Short Answer

Expert verified
The greatest common factor of 75 and 25 is 5.

Step by step solution

01

- Prime Factorization of 75

Find the prime factors of 75 by dividing it by the smallest prime numbers until all factors are prime. The prime factorization of 75 is:75 梅 3 = 2525 梅 5 = 55 梅 5 = 1So, the prime factors of 75 are 3 and 5.
02

- Prime Factorization of 25

Find the prime factors of 25 by dividing it by the smallest prime numbers until all factors are prime. The prime factorization of 25 is:25 梅 5 = 55 梅 5 = 1So, the prime factors of 25 are only 5.
03

- Identifying Common Factors

List the prime factors of both numbers:- Prime factors of 75: 3, 5- Prime factors of 25: 5Identify the common prime factors. The common prime factor is 5.
04

- Greatest Common Factor

Multiply all the common prime factors. Since the common prime factor is only 5, the greatest common factor (GCF) of 75 and 25 is 5.

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Key Concepts

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

prime factorization
Prime factorization involves breaking down a composite number into its prime numbers. Prime numbers are those that have only two divisors: 1 and themselves.
For example, in the provided exercise, we needed to find the prime factors of 75 and 25.

- **For 75:** we divide by the smallest prime (3) to get 25, and then divide 25 by 5 twice. This breaks down to: 75 = 3 脳 5 脳 5

- **For 25:** we divide by 5 twice and get: 25 = 5 脳 5

So, the prime factorization of 75 is 3 脳 5 脳 5, while for 25, it is 5 脳 5. This method helps us easily identify all prime factors connected to a number. Identifying prime factors is an essential step for finding the greatest common factor later.
common factors
Common factors are those that appear in the prime factorizations of multiple numbers. They help us determine shared divisors, key for finding the Greatest Common Factor (GCF).

To find common factors:

1. List all prime factors of each number. For instance:
- Prime factors of 75: 3, 5, 5
- Prime factors of 25: 5, 5
2. Identify which factors the numbers share. From the example:
- Both 75 and 25 share the factor: 5

Common factors are critical. They simplify calculations and reveal relationships between numbers. By determining common factors, recognizing the GCF becomes straightforward.
GCF calculation
The Greatest Common Factor (GCF) is the largest factor shared by two or more numbers. Once you've found the prime factors and common factors, calculating the GCF is simple.

Here is a quick way to find the GCF:

1. Identify all common factors:
- From our exercise, common factor for 75 and 25 is 5.
2. Multiply these common factors together to get the GCF:
- Since there's only one common factor (5), there's no need to multiply by anything else.

Therefore, the GCF of 75 and 25 is 5.

Using this method, you can determine the GCF for any set of numbers, easing your way through many math problems. Understanding the GCF simplifies addition, subtraction, and simplification of fractions.

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Most popular questions from this chapter

Subtract the polynomials. $$\left(-19 a^{5}-6 a^{3}+2 a^{2}+7\right)-\left(4 a^{5}+2 a^{3}+6 a^{2}-3\right)$$

Subtract the polynomials. $$\begin{array}{r}15 a^{3}-2 a^{2}+4 a \\\\-\left(2 a^{3}+6 a^{2}-12 a\right) \\\\\hline\end{array}$$

Simplify. Write the answers with positive exponents only. $$8^{2} 8^{-4}$$

Write a trinomial with degree 2.

Simplify. Write the answers with positive exponents only. $$5^{4} 5^{-7}$$
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