What Are The Common Factors Of 36 And 90

What Are the Common Factors of 36 and 90? A Deep Dive into Number Theory

Finding the common factors of two numbers might seem like a simple arithmetic exercise, but it's a fundamental concept in number theory with applications across various fields, from cryptography to computer science. This article will delve into finding the common factors of 36 and 90, explaining the process step-by-step and exploring related concepts. We'll also look at how to efficiently find common factors for larger numbers and discuss the importance of this concept in higher mathematics.

Understanding Factors

Before we tackle the specific problem of finding the common factors of 36 and 90, let's establish a clear understanding of what factors are.

A factor (also called a divisor) of a number is a whole number that divides the number exactly without leaving a remainder. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 evenly.

Finding Factors of 36

To find the factors of 36, we can systematically check each whole number from 1 up to 36:

• 1 x 36 = 36
• 2 x 18 = 36
• 3 x 12 = 36
• 4 x 9 = 36
• 6 x 6 = 36

Therefore, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Finding Factors of 90

Similarly, let's find the factors of 90:

• 1 x 90 = 90
• 2 x 45 = 90
• 3 x 30 = 90
• 5 x 18 = 90
• 6 x 15 = 90
• 9 x 10 = 90

Therefore, the factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

Identifying Common Factors

Now that we have the complete list of factors for both 36 and 90, we can identify the common factors. These are the numbers that appear in both lists. Comparing the two lists, we find the following common factors:

• 1
• 2
• 3
• 6
• 9
• 18

Therefore, the common factors of 36 and 90 are 1, 2, 3, 6, 9, and 18.

Greatest Common Factor (GCF)

Among the common factors, the largest one is called the Greatest Common Factor (GCF) or Highest Common Factor (HCF). In this case, the GCF of 36 and 90 is 18.

The GCF is a crucial concept in simplifying fractions and solving various mathematical problems. For example, to simplify the fraction 36/90, we can divide both the numerator and the denominator by their GCF (18), resulting in the simplified fraction 2/5.

Methods for Finding the GCF

While listing all factors works well for smaller numbers, it becomes inefficient for larger numbers. Here are some more efficient methods for finding the GCF:

1. Prime Factorization Method

This method involves finding the prime factorization of each number and then identifying the common prime factors.

• Prime factorization of 36: 2² x 3²
• Prime factorization of 90: 2 x 3² x 5

The common prime factors are 2 and 3². Multiplying these together gives us 2 x 3² = 2 x 9 = 18, which is the GCF.

2. Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the GCF of two numbers. It's based on the principle that the GCF of two numbers doesn't change if the larger number is replaced by its difference with the smaller number. This process is repeated until the two numbers are equal, which is the GCF.

Let's apply the Euclidean algorithm to 36 and 90:

1. 90 = 2 x 36 + 18
2. 36 = 2 x 18 + 0

The last non-zero remainder is 18, which is the GCF.

Applications of Common Factors and GCF

The concept of common factors and the GCF has numerous applications in various fields:

• Simplifying Fractions: As mentioned earlier, finding the GCF is essential for simplifying fractions to their lowest terms.

• Solving Equations: The GCF can be used to solve certain types of algebraic equations.

• Geometry: The GCF is used in geometric problems involving finding the greatest common divisor of lengths or areas.

• Cryptography: The concept of GCF plays a crucial role in RSA cryptography, a widely used public-key cryptosystem.

• Computer Science: The GCF is utilized in various algorithms and data structures in computer science, including the computation of least common multiples (LCM).

• Music Theory: The GCF is applied in music theory to determine the greatest common divisor of two musical intervals.

Least Common Multiple (LCM)

While we've focused on the GCF, it's also important to understand the concept of the Least Common Multiple (LCM). The LCM of two numbers is the smallest number that is a multiple of both numbers.

The relationship between the GCF and LCM is given by the formula:

LCM(a, b) x GCF(a, b) = a x b

For 36 and 90:

LCM(36, 90) x GCF(36, 90) = 36 x 90

LCM(36, 90) x 18 = 3240

LCM(36, 90) = 3240 / 18 = 180

Therefore, the LCM of 36 and 90 is 180. The LCM is essential when dealing with fractions, finding common denominators, and solving problems involving cycles or periodic events.

Conclusion

Finding the common factors of two numbers, like 36 and 90, is a fundamental skill in mathematics with far-reaching implications. Understanding this concept, along with the GCF and LCM, is crucial for mastering various mathematical topics and solving problems across different fields. Whether you are simplifying fractions, solving equations, or exploring more advanced mathematical concepts, the principles discussed in this article provide a strong foundation for your mathematical journey. The methods discussed, from simple listing to the efficient Euclidean algorithm, offer various approaches depending on the complexity of the numbers involved. Remember that a solid grasp of these concepts opens doors to a deeper understanding of number theory and its applications in the world around us.