What are the factors of 120

General
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what are the factors of 120

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What are the factors of 120?

Answer:

To determine the factors of 120, we need to identify all whole numbers that can divide 120 without leaving a remainder. Factors are integral components that, when multiplied together in pairs, result in the original number. Let’s explore the factors of 120 step by step, ensuring a comprehensive understanding of how to find them.

Step 1: Understand the Concept of Factors

A factor is a number that divides another number completely without leaving any remainder. For example, in the division (120 \div 10), because it results in a whole number (12), both 10 and 12 are factors of 120.

Step 2: Finding the Factors of 120

To find the factors, start by performing simple division starting with the smallest integers and check divisibility:

  1. 1: Any number is divisible by 1. So, 1 is a factor.
  2. 2: Since 120 is even, it is divisible by 2.
  3. 3: A quick divisibility test for 3 is to sum the digits (1 + 2 + 0 = 3), which is divisible by 3, so 120 is divisible by 3.
  4. 4: If the last two digits of a number are divisible by 4, then so is the number. 20 is divisible by 4, therefore 120 is also divisible by 4.
  5. 5: Numbers ending in 0 or 5 are divisible by 5. Hence, 120 is divisible by 5.
  6. 6: Since 120 is divisible by both 2 and 3, it must also be divisible by 6.
  7. 8: To check this, since 120 ends in 20, check if 20 is divisible by 8, and it is not. However, when we divide 120 by 8, the result is 15, which means 8 is a factor.
  8. 10: Ends in 0, so it’s divisible by 10.
  9. 12: Divide 120 by 12, the result is 10—a whole number, hence, 12 is also a factor.

Continue identifying factors until you reach numbers whose square exceeds 120:

Step 3: List All Factors

Now let’s see these operations together and also add complementary factors up to the point where they mirror each other:

  • (120 \div 1 = 120)
  • (120 \div 2 = 60)
  • (120 \div 3 = 40)
  • (120 \div 4 = 30)
  • (120 \div 5 = 24)
  • (120 \div 6 = 20)
  • (120 \div 8 = 15)
  • (120 \div 10 = 12)
  • (120 \div 12 = 10)
  • (120 \div 15 = 8)
  • (120 \div 20 = 6)
  • (120 \div 24 = 5)
  • (120 \div 30 = 4)
  • (120 \div 40 = 3)
  • (120 \div 60 = 2)
  • (120 \div 120 = 1)

Full Factor List

Hence, the complete list of factors of 120, in ascending order, is:
[ 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 ]

Visualization: Factor Pair Table

Here’s a simple table illustrating factor pairs, where two numbers multiply to result in 120.

Factor 1 Factor 2 Product
1 120 120
2 60 120
3 40 120
4 30 120
5 24 120
6 20 120
8 15 120
10 12 120

Prime Factorization

For completeness, you might want to consider the prime factorization of 120 for more detailed insights.

  1. Divide by 2: 120 ÷ 2 = 60
  2. Divide by 2: 60 ÷ 2 = 30
  3. Divide by 2: 30 ÷ 2 = 15
  4. Divide by 3: 15 ÷ 3 = 5
  5. Divide by 5: 5 ÷ 5 = 1

This shows the prime factorization of 120 is:
[ 120 = 2^3 \times 3 \times 5 ]

The use of prime factorization can also help derive the total number of factors through its formula. If you have 120 = 2^3 \times 3^1 \times 5^1, to find the total number of factors, apply (3+1)(1+1)(1+1) = 16, which matches the factors list provided.

This comprehensive breakdown was designed to ensure you understand all components related to the factors of 120. If you have any more questions or need further clarification, feel free to ask!