The greatest common factor (GCF) of two numbers is the largest number that both numbers share as a common factor. It is an important concept in mathematics, particularly in algebra and arithmetic. In this article, we will explore how to use the greatest common factor to find the largest shared factor between two numbers, in this case 36 and 90. We will also cover how to calculate the factors of both 36 and 90, and how to use those factors to discover the GCF.

Understanding the Greatest Common Factor

When finding the greatest common factor of two numbers, we are looking for the largest number that both of the numbers can evenly divide into without a remainder. Let’s take a look at the example of 36 and 90: 36 is a multiple of 6, and 90 is a multiple of 10. Since 6 is the largest number that both 36 and 90 can divide evenly into, 6 is the greatest common factor of 36 and 90.

The greatest common factor can also be used to simplify fractions. For example, if you have the fraction 8/24, you can divide both the numerator and denominator by the greatest common factor of 8 to simplify the fraction to 1/3.

Calculating the Greatest Common Factor

The first step to finding the greatest common factor of two numbers is to identify the factors of each number. Factors are all the numbers that can evenly divide into a given number. For example, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, and 45.

Once the factors of each number have been identified, the next step is to compare the two lists and find the greatest number that appears in both lists. This number is the greatest common factor of the two numbers. In the example above, the greatest common factor of 36 and 90 is 18.

Factors of 36

The first number we are looking at is 36. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

It is important to note that the factors of 36 are all whole numbers. This means that the factors of 36 cannot be fractions or decimals. Additionally, the factors of 36 are all integers, meaning that they are all positive or negative whole numbers.

Factors of 90

The second number we are looking at is 90. The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, and 45.

It is important to note that the factors of 90 are all whole numbers. This means that the factors of 90 cannot be fractions or decimals. Additionally, the factors of 90 are all integers, meaning that they are all positive or negative whole numbers.

Prime Factors of 36 and 90

The next step is to break down each number into its prime factors. The prime factors of 36 are 2 × 2 × 3 × 3. The prime factors of 90 are 2 × 3 × 3 × 5. When factoring one number using its prime factors, we use only factors that are common to that number and the other one we’re comparing it against.

For example, the common factors of 36 and 90 are 2, 3, and 5. We can use these common factors to simplify the equation and find the greatest common factor (GCF). The GCF of 36 and 90 is 18, which is the product of the common factors (2 × 3 × 3).

Finding the Greatest Common Factor

Once we have established all the common prime factors between 36 and 90, we can find their greatest common factor (GCF). To calculate the GCF, we simply multiply the common factors together. In this case, the common factors are 2 × 3 × 3 × 5. When we multiply them all together, we get 90. The greatest common factor of 36 and 90 is therefore 90.

It is important to note that the greatest common factor is not always the same as the least common multiple (LCM). The LCM is the smallest number that is a multiple of both numbers. In this case, the LCM of 36 and 90 is 180. This is because 180 is the smallest number that is a multiple of both 36 and 90.

The End Result

In conclusion, when two numbers such as 36 and 90 are being compared for their greatest common factor, it is important to identify all their factors and then select only those which are common to both. Once we have this list of common factors, all we need to do is multiply them together to get the GCF. In this case, the greatest common factor of 36 and 90 was determined to be 90.