Factors.

Factors are very helpful in solving all kinds of math problems. Mentioned below are some important factors in math.

• Using factors, we can break down big numbers into smaller parts to solve them easily.
   
• They help find patterns in math by showing how numbers are connected. 
   
• Factors are used in real-life scenarios like sports, cooking, gardening, etc.
   

Factors play an important role in algebra and arithmetic. Here are some of the key topics we learned about factors.

• Prime Factors
• Common Factors
• Greatest Common Factors
   

Prime factors are those factors that are prime in nature. Therefore, these factors can only be divisible by 1 and the original number itself. Prime factorization is a process of breaking down a given number into its prime factors. It is like breaking down a LEGO creation into its basic parts. Prime factors can be used to form groups, find patterns, etc. 

The prime factors of 12 are 2 and 3.

Common factors are those numbers that can be considered as factors of two or more numbers. They are helpful when we solve problems that involve multiple numbers. We use common factors while sharing resources, grouping, etc.

Let’s understand common factors with an example. 1, 2, 3, and 6 are the common factors of 12 and 18. This can be determined by identifying the factors of 12 and 18 separately. While the factors of 12 are 1, 2, 3, 4, 6, 12, the factors of 18 are 1, 2, 3, 6, 9, 18. Now, among these factors, only 1, 2, 3, and 6 can be seen in both the lists. Hence, they are the common factors.  
 

The greatest common factor or GCF is the largest common factor of two or more numbers. It is also called the greatest common divisor (GCD). GCF helps in simplifying problems like reducing fractions or dividing items into groups. 

Let’s say we need to determine the GCF of 12 and 18. To find the GCF, we should first find the factors of 12 and 18. Then, we can simply choose the greatest number among the common factors. 

Factors of 12: 1, 2, 3, 4, 6, 12.

Factors of 18: 1, 2, 3, 6, 9, 18.

So, GCF of 12 and 18 is 6.
 

Factors can be used in many fields from everyday tasks to advanced problem-solving etc. Let's explore some of their practical uses with examples.

Distributing items equally: Factors help with fair distribution when dividing items into groups without leaving any leftovers. For instance, it can be used in a situation where 12 candies need to be distributed equally to 6 children. This problem can be solved by dividing 12 by 6. Since 12/6 is 2, each child will get 2 candies.

Simplifying fractions: GCFs are used to simplify the fractions to their lowest terms. For example, simplify fractions 1824 . 

Factors of 18 are 1, 2, 3, 6, 9, and 18.

Factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Common factors of 18 and 24 are 1, 2, 3, and 6.

 GCF is the largest common factor.

Since 6 is the largest number among the common factors of 18 and 24, the GCF is 6.

Divide the numerator and denominator by 6.

1824 = 18/624/6 = 34 

Finding patterns in sequences: Factors can be used to identify intervals in sequences or patterns that repeat. For e.g., if we need to plan one particular event in the next 2 days and a second event in 6 days' time, we can use LCM to determine the merging point of the events. Since the LCM of 2 and 6 is 6, the events will merge every 6 days.
 

To learn factors more quickly, use the following tips and tricks.

• Instead of listing all the factors, find the pairs of the factors that help in finding the factors easily.
   
• Breaking the number into prime factors helps in solving many factor-related problems, like finding the LCM, GCF, etc.
   
• Begin with 1 and test numbers successively up to the square root of the number, checking which numbers divide it evenly. Every time we find one, we should pair it with its matching factor.