**Factors of 60** are **1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60**.

Here is complete detailed methods to find the factors of 60.

Actually, Factor is a number which divides any number completely without leaving remainder.

Or simply we can say that if we multiply two whole numbers (positive & negative) and it gives a product. Then the numbers that we multiply, are the factors of the product.

For example, if we multiply 3 to 20, it gives 60. Here 3 and 20 are called positive factors and if we multiply -4 to -15, it gives 60. So -4 and -15 are called negative factors.

Table of Contents

## How to Find The Factors of 60 ?

Since factors of 60 are the numbers, which divides 60 completely without leaving remainder. So here we are going to divide 60 by all numbers up to 60. Let’s see the process –

60 Divided by | Result | Product (Col 'A' x 'B') | Remark |
---|---|---|---|

Column 'A' | Column 'B' | Column 'C' | Column 'D' |

1 | 60 | 60 | Accepted |

2 | 30 | 60 | Accepted |

3 | 20 | 60 | Accepted |

4 | 15 | 60 | Accepted |

5 | 12 | 60 | Accepted |

6 | 10 | 60 | Accepted |

7 | 8.571... | 60 | Rejected |

8 | 7.5 | 60 | Rejected |

9 | 6.66... | 60 | Rejected |

10 | Stop the process here, because 10 has already came in column 'B' |

We have to stop the process, if any number of column ‘B’ appears in column ‘A’.

Here we stop the division when 10 appears in column ‘A’, which has already been came in column ‘B’.

Take another example, Factors of 36

In above example of factors of 36, no element of Column ‘B’ appears in Column ‘A’, but we stopped the process when 6 comes, because on dividing by 6, it produces the same number (6 = 6) .

So we have to stop the process when any element comes in both columns at same level and this element will consider as a factor.

Hence, all the factor of 60 in above table 1 are : 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Similarly, negative factor of 60 are : -1, -2, -3, -4, -5, -6, -10, -12, -15, -20, -30, -60

## What Are The Factors of 60 in Pairs ?

As we saw in above table, 1 multiply by 60 gives 60. So (1, 60) or (60, 1) is a factor pair of 60. Let us see how many factor pairs can be achieved –

1 x 60 = 60

2 x 30 = 60

3 x 20 = 60

4 x 15 = 60

5 x 12 = 60

6 x 10 = 60

10 x 6 = 60

12 x 5 = 60

15 x 4 = 60

20 x 3 = 60

30 x 2 = 60

60 x 1 = 60

Similarly, negative factor pairs of 60 can be written as follows –

-1 x -60 = 60

-2 x -30 = 60

-3 x -20 = 60

-4 x -15 = 60

-5 x -12 = 60

-6 x -10 = 60

-10 x -6 = 60

-12 x -5 = 60

-15 x -4 = 60

-20 x -3 = 60

-30 x -2 = 60

-60 x -1 = 60

Hence, 60 has 12 factor pairs.

## What Are the Prime Factors of 60 ?

The **Prime Factors of 60 **are **2, 3, 5**. Here we will discuss, mainly two methods of factorization to find Prime Factors of any number.

- Division Method
- Factor Tree Method

### Prime Factors of 60 by Division Method

**Step 1 :** Divide the number by the smallest prime number, which divides the number exactly.

**Step 2 :** Divide the quotient again by the smallest prime number. If quotient is not exactly divisible by the

smallest prime number, choose next smallest prime number.

**Step 3 :** Repeat Step 2 again and again till the quotient becomes 1.

**Step 4 :** All the prime numbers used in above process are the prime factors.

Check : Multiply all the prime factors, the multiplication should be the number itself.

Caution : In this method, only prime numbers will be use to divide.

#### Steps to Find Prime Factors of 60

- First of all, we divide 60 by smallest prime number i.e. 2 and we got 30 as quotient and remainder is zero. (Note – if remainder will not zero, means doesn’t divide exactly, try next prime number to divide)
- Again we divide the quotient 30 by smallest prime number i.e. 2 and we got 15 as quotient. Again we divide the quotient 15 by smallest prime number i.e. 2, but 2 doesn’t divide 15 exactly. So we try to divide next prime number i.e. 3
- Repeat step 2 until we got 1 as a quotient.
- Here 2 used five times.

Hence, 60 = 2 x 2 x 3 x 5 = 2^{2} x 3^{ }x 5

So **60 has three prime factor i.e. 2, 3 & 5**

If we take another example prime factors of 32 = 2^{5}

So, 32 has only one prime factor i.e. 2.

### Factor Tree of 60

**Step 1 :** Write a number as root of the tree

**Step 2 :** Find the factors of the number, each factor will be considered as root again (now, we got 2 new root)

**Step 3 :** Repeat step 2 until we can’t factor any more (or until we got prime factor)

**Step 4 :** The end nodes are the prime factors.

We can initiate the tree by choosing any two factors. Here we choose 6 and 10 as factors of 60 in first tree, 3 and 20 in second tree and 2 & 30 in third tree.

We got the same unique nodes in every tree.

The Factor Tree may be made so many types depends upon choosing the initial and middle factors.

## Number of Factors of 60

The **Number of Factors of 60 is 12**. To find the number of factors, we use Division Method.

In our example, as we saw above 60 = 2^{2} x 3^{ }x 5

**Number of factors is = (power + 1)(power + 1)(power + 1)………. **

Hence number of **factors of 60** is= (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 =12

and if we consider both positive and negative factors, then

total no. of factors = 12 (positive) + 12 (negative) = 24

I hope, this step-by-step tutorial will be helpful to understand factorization method of 60.

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