Factors of 60: Definition, Pairs, and Prime Factorization

Factors of 60: Factors are the integers that divide a number exactly, leaving no remainder. The factors of 60 are all the numbers that can be multiplied in pairs to get 60. Here, we will learn to find the pair factors and prime factorization of 60 with solved examples to help students to understand the concept easily.

Read More: Prime Factorization Method 

What Are Factors of 60?

In mathematics, a factor of a number is an integer that divides the number exactly without leaving a remainder.

The factors of 60 are all numbers that divide 60 completely. In other words, they are the divisors for the dividend 60 without leaving any remainders. Therefore, we start dividing 60 by numbers sequentially:

60 ÷ 1 = 60

60 ÷ 2 = 30

60 ÷ 3 = 20

60 ÷ 4 = 15

60 ÷ 5 = 12

60 ÷ 6 = 10

60 ÷ 10 = 6

60 ÷ 12 = 5

60 ÷ 15 = 4

60 ÷ 20 = 3

60 ÷ 30 = 2

60 ÷ 60 = 1

So, we can say that the factors of 60 are

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

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Prime Factorization of 60

One of the most common methods for finding the factors of a number is prime factorization. It is the method of expressing a number as the product of prime numbers.

Let’s see how to carry out the prime factorization of 60 step by step:

• Divide 60 by the smallest prime number, 2, and get 30 as a result

• Divide 30 again by 2 and get 15 as a result.

• 15 is not divisible by 2, but it is divisible by the next prime number, 3. By dividing 15 by 3, we get 5.

• We divide 5 by 5 and get 1 as a result.

So, 60 can be expressed as the product of prime factors as follows: 60 = 2 x 2 x 3 x 5. Therefore, the prime factors of 60 are 2, 3, and 5.

Read More: Factors of 36

Pair Factors of 60

An interesting point about the factors is that you can group them in pairs so that each pair gives a product equal to the given number. These are called pair factors. We can find the pair factors of 60 so that each pair gives a product of 60. The table below shows how you get the pair factors of 60:

So, the pair factors of 60 are: (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), and (6, 10).

These pairs are very useful in solving factor-related problems, such as finding least common multiples (LCM).

Read More:  What is the Factorial of 100?

Factor Tree of 60

Factor Tree refers to the diagrammatic representation of the factors of a number, resembling the branches of trees. It can be done by step-wise division of the number 60 until we reach the prime factors. Here is a step-wise procedure to carry out the process:

Step 1: Split 60 into two factors with 2 and 30 as two branches.

Step 2: Keeping 2 as the factor, factorize 30 with 2 and 15 as two factors, represented by two branches.

Step 3: In the next step, represent 3 and 5 as two prime factors of 15.

Step 4: Arrange the prime factors at the end of each branch in ascending order.

Thus, we get the prime factorisation of 60 = 2 × 2 × 3 × 5

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Negative Factors of 60

Factors can also be negative because when two negative factors are multiplied, it gives a positive value.  Let’s look at the negative factors of 60 as given below:

So, the negative factors of 60 are:

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

Similarly, the negative pair factors of 60 are:

(-1, -60), (-2, -30), (-3, -20), (-4, -15), (-5, -12), and (-6, -10).

Easy Tricks to Find factors of 60

There are some easy ways to find the factors of 60 as explained below:

• The number 60 is an even number, so it is divisible by 2.

• Since 60 ends with zero, it must be divisible by 5 and 10.

• It is divisible by 6 and hence also divisible by 2 and 3.

• Since it is divisible by 3 and 5, it must have 15 as a factor.

• The number is divisible by 2 and 6, so it must have 12 as a factor.

Also read: What is Factorization Formula?

Real Life Applications of Factors of 60

Factors are not only useful in academics, but they also find a place in many real-life scenarios, as mentioned below:

• Distribution among groups: If your child wants to distribute 60 chocolates equally among friends, factors of 60 help them determine possible group sizes (such as 2, 3, 4, 5, 6, 10, 20, etc.).

• Time Measurement: An hour means 60 minutes. Factors of 60 can help your child divide time evenly for different activities.

• Fractions and Ratios: Factors are the most useful tool to simplify fractions. For example, 16/36 can be simplified as 4/9 by dividing the numerator and denominator by common factors 2 and 3.

• Geometry and Patterns: Factor pairs become useful when arranging a certain number of objects in rows and columns. For example, 60 flowerpots can be arranged in 5 rows with 12 pots in each row.

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