Common Factors

When a number is divided exactly by another number, that divisor is known as a factor of the original number. A factor is always less than or equal to the number itself, with 1 being a factor of every number. Similarly, each number is also its own factor; if a number is a multiple of a divisor, it can be divided evenly by that divisor and possibly by several others as well. Common factors are numbers that are shared by two or more numbers. That is, a common factor is any number that divides each of the numbers exactly, leaving no remainder. 

For example, 

The common factors of 36 and 18 can be found by listing the factors of each number.

Factors of 36 are:

Let us factor 36. 

1, 2, 3, 4, 6, 9, 12, 18, and 36.

Factors of 18 are:

Let us factor 18.

1, 2, 3, 6, 9, and 18.

The common factors of 18 and 36 are the factors that are common to both 18 and 36. Here, 1, 2, 3, 6, 9, and 18 are the numbers that are common to both 18 and 36.

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

Now, let’s learn how to find the common factors of any two or more numbers. Follow these steps to find the common factors of any two or more numbers.
 

Step 1: List all the factors of the given numbers

Step 2: Identify all the common numbers from the list; these are the common factors of the numbers. 

Let us understand this concept with the help of a few examples. 

Common factors of 5 and 16.

Factors of 5 are 1 and 5.

Factors of 16 are 1, 2, 4, 8, and 16.

Therefore, the common factors of 5 and 16 are 1.

Common factors of 27 and 60.

Factors of 27 are 1, 3, 9, and 27.

Factors of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

Therefore, the common factors of 27 and 60 are 1 and 3.

Common factors of 15 and 25.

Factors of 15 are 1, 3, 5, and 15.

Factors of 25 are 1, 5, and 25.

Therefore, the common factors of 15 and 25 are 1 and 5.

Common factors of 45 and 65

Step 1: Listing the factors of 45 and 65

The factors of 45: 1, 3, 5, 9, 15, 45

The factors of 65: 1, 5, 13, 65

Step 2: Here, the common factors of 45 and 65 are 1 and 5.
 

The largest number from the common factors of the numbers is the GCF of the numbers. GCF stands for greatest common factor. The GCF of two numbers is the largest number that divides both evenly. For example, if \(\text{GCF(a, b) = x,}\) then x divides both a and b evenly. Here, a and b are the given numbers, and x is their greatest common factor.

For instance, let’s find the GCF of 45 and 65 
 

The factors of 45: 1, 3, 5, 9, 15, 45

The factors of 65: 1, 5, 13, 65

Here, the common factors are 1 and 5

So, the GCF of 45 and 65 is 5.

Least Common Factor:

The least common factor, on the other hand, is the smallest positive common factor of two or more numbers. It is the opposite of GCF. While there may be multiple common factors among two or more numbers, there can be only one GCF and one LCF.

For example, let us find the least common multiple of 54 and 72.

Factors of 54 are: 

1, 2, 3, 6, 18, 27, and 54.

Factors of 72 are:

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

Therefore, the common factors of 54 and 72 are 1, 2, 3, 6, and 18.

Hence, the least common factor of 54 and 72 is 1.

Identifying common factors quickly is important in math and everyday situations, like dividing, sharing, or simplifying numbers. These tips and tricks that help to make it easy to find them without lengthy calculations.

• Memorize the small number pattern to save time.
   
• Break numbers into smaller parts to make calculation easily.
   
• Write all factors of each number, then find the common ones.
   
• Always check carefully to avoid mistakes when listing or multiplying factors.
   
• Teachers can start teaching the lesson by asking students some interactive questions, like “We have 12 apples. In how many equal groups can we share them?” This helps students understand common factors on their own.
   
• Parents can use some factors, such as rainbows, to identify the factors very easily. Ask your children to write a number and draw rainbow arcs connecting the factor pairs. This helps kids see pairs rather than memorize them.
   
• Teachers can help students build an understanding through multiples. Show them the connection between factors and multiples. While factors go into a number, the multiples come out of a number. We can help learners see both, where common factors become easier to spot.
   
• Encourage the learners to break numbers into primes using factor trees. It becomes easier for kids to compare the common primes this way.

The factors that are common among two or more numbers are the common factors of any number. In this section, let’s see how and why we use the common factors in real-life situations. 
 

• Simplifying Fractions: In mathematics, we use common factors to find the GCF of any two or more numbers. 
   
• Finding Common Denominators: When doing the basic operations in fractions, we use common factors to simplify the fraction and to find the common denominator.
   
• Resource Allocation: For resource allocation, scaling models, electric circuits, manufacturing and production, etc., we use common factors in the field of engineering. 
   
• Cooking: To adjust the quantities of the recipe, we use common factors in cooking.
   
• Music and Rhythm: In music theory, the common factors can help to synchronize rhythms by finding patterns that repeat evenly, allowing different beats or instruments to align periodically.