What is Factors of 120?

Factors of 120?

Answer:

The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120.

Method: Prime Factorization

To find the factors of 120, we can use the method of prime factorization. This involves breaking down the number into its prime factors, which are the prime numbers that multiply together to give the original number.

Step-by-Step Calculation:

1. Find the smallest prime factor: The smallest prime factor of 120 is 2 because 120 is an even number.

   [ 120 \div 2 = 60 ]

2. Continue with prime factorization: Repeat the process with the quotient (60 in this case), which is also even.

   [ 60 \div 2 = 30 ]

3. Repeat the process: Continue factoring until you cannot divide by 2 anymore.

   [ 30 \div 2 = 15 ]

   Now, 15 is not divisible by 2, so we move to the next smallest prime number, which is 3.

4. Divide by the next smallest prime number: Divide 15 by 3.

   [ 15 \div 3 = 5 ]

5. Identify the last prime factor: The number 5 is a prime number, so we have now found all the prime factors of 120.

   The prime factorization of 120 is:

   [ 120 = 2^3 \times 3 \times 5 ]

List of Factors:

Now, to find all the factors of 120, we combine the prime factors in different ways:

• Multiplying 2 by itself and by other prime factors:

  • ( 2 \times 1 = 2 )
  • ( 2 \times 2 = 4 )
  • ( 2 \times 3 = 6 )
  • ( 2 \times 2 \times 2 = 8 )
  • ( 2 \times 5 = 10 )
  • ( 2 \times 2 \times 3 = 12 )
  • ( 2 \times 2 \times 5 = 20 )
  • ( 2 \times 3 \times 5 = 30 )
  • ( 2 \times 2 \times 2 \times 3 = 24 )
  • ( 2 \times 2 \times 2 \times 5 = 40 )
  • ( 2 \times 2 \times 3 \times 5 = 60 )

• Multiplying 3 by itself and by other prime factors:

  • ( 3 \times 1 = 3 )
  • ( 3 \times 5 = 15 )

• Multiplying 5 by itself and by other prime factors:

  • ( 5 \times 1 = 5 )

• And finally, the number itself:

  • ( 120 )

Verification:

To ensure that we have all the factors, we can check that each factor pairs with another to give the product of 120. For example, ( 2 \times 60 = 120 ), ( 3 \times 40 = 120 ), and so on.

Related Knowledge Points:

• Prime Numbers: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
• Prime Factorization: The process of breaking down a composite number into its prime factors.
• Factors: Factors of a number are the numbers that divide into it exactly without leaving a remainder.

Detailed Explanation:

Factors are the building blocks of a number. They are important in various branches of mathematics, including number theory and algebra. Understanding how to find the factors of a number through prime factorization is a fundamental skill that helps in simplifying fractions, finding greatest common divisors, and solving various mathematical problems.

In the case of 120, by breaking it down into its prime factors, we can systematically determine all possible combinations that give us the original number when multiplied together. This method ensures that we do not miss any factors and provides a clear and organized approach to finding all factors of a given number.