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# What are all the factors, the prime factorization, and factor pairs of 60?

To find the factors of 60, divide 60 by each number starting with 1 and working up to 60

Now let us find how to calculate all the factors of 60:
2
= 30
3
= 20
4
= 15
5
= 12
6
= 10
As you can see, the factors of 60 are:
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

## Introduction to Factors

Factors are the building blocks of numbers, referring to the whole numbers we can multiply together to produce a given product. With 60 as an illustrative example, uncovering its factors can provide a deeper understanding of its mathematical characteristics and applications.

## What are Factors of 60?

The factors of 60 are the whole numbers that, when multiplied in pairs, equal 60. This concept is foundational in understanding how we can break down and analyze numbers in a variety of mathematical contexts. Below is a list of factors for 60:

• 1
• 2
• 3
• 4
• 5
• 6
• 10
• 12
• 15
• 20
• 30
• 60

We can visualize these factors and their pairings: – 1 × 60 = 60 – 2 × 30 = 60 – 3 × 20 = 60 – 4 × 15 = 60 – 5 × 12 = 60 – 6 × 10 = 60

## How to Find the Factors of 60?

To find the factors of 60, we use the division method. This process involves dividing 60 by integers that result in whole numbers, indicating they are factors. Here’s an illustrative example:

1. Start with the number 1; since all numbers have 1 as a factor: 60 ÷ 1 = 60
2. Test the next whole number: 60 ÷ 2 = 30, so 2 is a factor
3. Continue testing integers: 60 ÷ 3 = 20, making 3 a factor
4. Proceed sequentially: 60 ÷ 4 = 15, confirming that 4 is a factor
5. And so on, until you reach a quotient equal to the divisor, which marks the end of possible factors for 60

Any number beyond this point will result in a quotient lower than the divisor, which signifies it is not a factor of 60. This is because factors of a number n will always be less than or equal to √n.

## Pair Factors of 60

Each factor of 60 has a complement with which it can be multiplied to give the product of 60. Here, both positive and negative pair factors are presented.

Positive Pair Factor Negative Pair Factor
(1, 60) (-1, -60)
(2, 30) (-2, -30)
(3, 20) (-3, -20)
(4, 15) (-4, -15)
(5, 12) (-5, -12)
(6, 10) (-6, -10)

## Prime Factorization of 60

To distill 60 into its prime components, we employ prime factorization, a technique revealing the prime number building blocks that multiply together to form 60. Below are the steps for the prime factorization of 60:

1. Divide 60 by its smallest prime factor: 60 ÷ 2 = 30
2. Continue dividing by 2 as long as the result is even: 30 ÷ 2 = 15
3. Since 15 is not divisible by 2, move on to the next smallest prime, which is 3: 15 ÷ 3 = 5
4. Finally, divide by 5, which is also a prime number: 5 ÷ 5 = 1
5. Now that we’ve reached 1, our prime factorization process is complete

The prime factorization of 60 can be expressed as 2 × 2 × 3 × 5 = 2² × 3 × 5, showcasing its prime factors: 2, 3, and 5.

## Important Points to Remember

• Total count of factors of 60: 12
• Prime factors of 60: 2, 3, 5
• Pair factors of 60: can be positive or negative, such as (1, 60) and (-1, -60)
• Sum of all factors of 60: 1 + 2 + 3 + 4 + 5 + 6 + 10 + 12 + 15 + 20 + 30 + 60 = 168

Through the analysis of 60’s factors, we explore its divisibility, witness the role of prime numbers in its composition, and gain insights into its usage in various mathematical scenarios. This investigation of 60’s factors expands our comprehension of how numbers can be deconstructed and interconnect.

## Exercises

1. Write down all the prime factors of 60.
2. Using the factor pairs of 60, find two numbers that their product equals 60.
3. Calculate the sum of all the factors of 60.

## Solutions to Exercises

1. The prime factors of 60 are 2, 3, and 5.
2. A pair of numbers whose product equals 60 are 6 and 10 (6 × 10 = 60).
3. The sum of all factors of 60 is 168 (1 + 2 + 3 + 4 + 5 + 6 + 10 + 12 + 15 + 20 + 30 + 60).

### What are the Factors of 60?

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

### What is the Greatest Factor of 60?

The greatest factor of 60 is the number itself, which is 60.

### Is 60 a Prime or Composite Number?

60 is a composite number because it has more factors other than 1 and itself.