LCM

# How To Find LCM of 80, 85, and 90? | Listing, Division, and Prime Factorization Method

Written by Prerit Jain

Updated on: 18 Feb 2023

Contents

### How To Find LCM of 80, 85, and 90? | Listing, Division, and Prime Factorization Method

LCM of 80, 85, and 90 is 12240. LCM of 80, 85, and 90, also known as Least Common Multiple or Lowest Common Multiple of 80, 85, and 90 is the lowest possible common number that is divisible by 80, 85, and 90.

Now, let’s see how to find the LCM of 80, 85, and 90.

Multiples of 80 are 80, 160, 240, 320, 400, 480, 560, 640, 720, 800,…, 12000, 12080, 12160, 12240, 12320, 12400,…

Multiples of 85 are 85, 170, 255, 340, 425, 510, 595, 680, 765, 850,…, 11985, 12070, 12155, 12240, 12325, 12410,…

Multiples of 90 are 90, 180, 270, 360, 450, 540, 630, 720, 810, 900,…, 11880, 11970, 12060, 12150, 12240, 12330, 12420,… Here, 12240 is the common number in the multiples of 80, 85, and 90, respectively, or that are divisible by 80, 85, and 90. So, 12240 is the lowest common number among all the multiples that is divisible by 80, 85, and 90, and hence the LCM of 80, 85, and 90 is 12240.

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## Methods to Find the LCM of 80, 85, and 90

There are three different methods for finding the LCM of 80, 85, and 90. They are:

1. Division Method

2. Listing Method

3. Prime Factorization Method

## LCM of 80, 85, and 90 Using the Division Method

The division method is one of the methods for finding the LCM. To find the LCM of 80, 85, and 90 using the division method, divide 80, 85, and 90 by the smallest prime number, which is divisible by any of them. Then, the prime factors further obtained will be used to calculate the final LCM of 80, 85, and 90.

Follow the following steps to find the LCM of 80, 85, and 90 using the division method:

**Step 1:**Write the numbers for which you have to find the LCM, that is 80, 85, and 90 in this case, separated by commas.**Step 2:**Now, find the smallest prime number, which is divisible by either 80 or 85, or 90.**Step 3:**If any of the numbers among 80, 85, and 90 are not divisible by the respective prime number, write that number in the next row just below it and proceed further.**Step 4:**Continue dividing the numbers obtained after each step by the prime numbers, until you get the result as 1 in the entire row.**Step 5:**Now, multiply all the prime numbers and the final result will be the LCM of 80, 85, and 90.

LCM of 80, 85, and 90 can be obtained using the division method:

Prime Factors | First Number | Second Number | Third Number |

2 | 80 | 85 | 90 |

2 | 40 | 85 | 45 |

2 | 20 | 85 | 45 |

2 | 10 | 85 | 45 |

3 | 5 | 85 | 45 |

3 | 5 | 85 | 15 |

5 | 5 | 85 | 5 |

17 | 1 | 17 | 1 |

1 | 1 | 1 |

So, the LCM of 80, 85, and 90 = 2 * 2 * 2 * 2 * 3 * 3 * 5 * 17= 12240

## LCM of 80, 85, and 90 Using the Listing Method

The listing method is one of the methods for finding the LCM. To find the LCM of 80, 85, and 90 using the listing method, follow the following steps:

**Step 1:**Write down the first few multiples of 80, 85, and 90 separately.**Step 2:**Out of all the multiples of 80, 85, and 90, focus on the multiples that are common to both the numbers, that is, 80, 85, and 90.**Step 3:**Now, out of all the common multiples, take out the smallest common multiple. That will be the LCM of 80, 85, and 90.

LCM of 80, 85, and 90 can be obtained using the listing method:

- Multiples of 80 are 80, 160, 240, 320, 400, 480, 560, 640, 720, 800,…, 12000, 12080, 12160, 12240, 12320, 12400,…
- Multiples of 85 are 85, 170, 255, 340, 425, 510, 595, 680, 765, 850,…, 11985, 12070, 12155, 12240, 12325, 12410,…
- Multiples of 90 are 90, 180, 270, 360, 450, 540, 630, 720, 810, 900,…, 11880, 11970, 12060, 12150, 12240, 12330, 12420,…

Here, it is clear that the least common multiple is 12240. So, the LCM of 80, 85, and 90 is 12240.

## LCM of 80, 85, and 90 Using the Prime Factorization Method

The prime factorization method is one of the methods for finding the LCM. To find the LCM of 80, 85, and 90 using the prime factorization method, follow the steps given below:

**Step 1:**Find the prime factors of 80, 85, and 90 using the repeated division method.**Step 2:**Write all the prime factors in their exponent forms. Then multiply the prime factors having the highest power.**Step 3:**The final result after multiplication will be the LCM of 80, 85, and 90.

LCM of 80, 85, and 90 can be obtained using the prime factorization method:

- Prime factorization of 80 can be expressed as 2 * 2 * 2 * 2 * 5 = 2
^{1}* 2^{1}* 2^{1 }* 2^{1 }* 5^{1 }= 2^{4}* 5^{1} - Prime factorization of 85 can be expressed as 5 * 17 = 5
^{1}* 17^{1} - Prime factorization of 90 can be expressed as 2 * 3 * 3 * 5 = 2
^{1}* 3^{1}* 3^{1}^{ }* 5^{1}= 2^{1}* 3^{2 }* 5^{1}

So, the LCM of 80, 85, and 90 = 2^{4} * 3^{2} * 5^{1} * 17^{1 }= 2 * 2 * 2 * 2 * 3 * 3 * 5 * 17 = 12240

## The Formula for Finding the LCM of 80, 85, and 90

LCM of 80, 85, and 90 can be calculated using the formula:

LCM (80, 85, 90) = [(80 * 85 * 90) * HCF (80, 85, 90)] / [HCF (80, 85) * HCF (85, 90) * HCF (80, 90)]

where HCF is the highest common factor or the greatest common divisor.

## Problems Based on LCM of 80, 85, and 90

**Q 1: Find the smallest number divisible by 80, 85, and 90.Solution:**

The smallest number divisible by 80, 85, and 90 is the LCM of 80, 85, and 90.

Using the listing method, we can find the LCM of 80, 85, and 90:

- Multiples of 80 are 80, 160, 240, 320, 400, 480, 560, 640, 720, 800,…,12000, 12080, 12160, 12240, 12320, 12400,…
- Multiples of 85 are 85, 170, 255, 340, 425, 510, 595, 680, 765, 850,…,11985, 12070, 12155, 12240, 12325, 12410,…
- Multiples of 90 are 90, 180, 270, 360, 450, 540, 630, 720, 810, 900,…,11880, 11970, 12060, 12150, 12240, 12330, 12420,…

Here, the smallest number divisible by 80, 85, and 90 is 12240.

**Q 2: Find the LCM of 80, 85, and 90 using the prime factorization method.Solution:**

To find the LCM of 80, 85, and 905 using the prime factorization method, first, we will find the prime factors of 80, 85, and 90 using the repeated division method. Then, we will write all the prime factors in their exponent forms and multiply the prime factors having the highest power. The final result after multiplication will be the LCM of 80, 85, and 90.

- Prime factorization of 80 can be expressed as 2 * 2 * 2 * 2 * 5 = 2
^{1}* 2^{1}* 2^{1 }* 2^{1 }* 5^{1 }= 2^{4}* 5^{1} - Prime factorization of 85 can be expressed as 5 * 17 = 5
^{1}* 17^{1} - Prime factorization of 90 can be expressed as 2 * 3 * 3 * 5 = 2
^{1}* 3^{1}* 3^{1}^{ }* 5^{1}= 2^{1}* 3^{2 }* 5^{1}

So, the LCM of 80, 85, and 90 = 2^{4} * 3^{2} * 5^{1} * 17^{1 }= 2 * 2 * 2 * 2 * 3 * 3 * 5 * 17 = 12240

**Q 3: What are the other numbers having the LCM as 12240? Show the representation using the division method.Solution:**

Other than 80, 85, and 90, LCM of 144 and 85 is also 12240. We will prove this using the division method.

To find the LCM of 80, 85, and 90 using the division method, first, we will find the smallest prime number, which is divisible by either 80 or 85 or 90. If any of the numbers among 80, 85, and 90 is not divisible by the respective prime number, we will write that number in the next row just below it and proceed further. We will continue dividing the numbers obtained after each step by the prime numbers until we get the result as 1 in the entire row. We will multiply all the prime numbers and the final result will be the LCM of 80, 85, and 90.

Prime Factors | First Number | Second Number | Third Number |

2 | 80 | 85 | 90 |

2 | 80 | 30 | 90 |

3 | 80 | 15 | 90 |

3 | 15 | 5 | 25 |

5 | 5 | 5 | 25 |

5 | 1 | 1 | 5 |

1 | 1 | 1 |

So, the LCM of 80, 85, and 90 = 2 * 2 * 3 * 3 * 5 * 5 = 12240

Now, to find the LCM of 144 and 85 using the division method, first, we will find the smallest prime number, which is divisible by either 144 or 8. If any of the numbers between 144 and 85 is not divisible by the respective prime number, we will write that number in the next row just below it and proceed further. We will continue dividing the numbers obtained after each step by the prime numbers until we get the result as 1 in the entire row. We will multiply all the prime numbers and the final result will be the LCM of 144 and 85.

Prime Factors | First Number | Second Number |

2 | 144 | 85 |

2 | 72 | 85 |

2 | 36 | 85 |

2 | 18 | 85 |

3 | 9 | 85 |

3 | 3 | 85 |

5 | 1 | 85 |

17 | 1 | 17 |

1 | 1 |

So, the LCM of 80, 85, and 90 = 2 * 2 * 2 * 2 * 3 * 3 * 5 * 17 = 12240

**Q 4: Find the LCM of 80, 85, and 90 using the listing method.Solution:**

To find the LCM of 80, 85, and 90 using the listing method, first, we will write down the first few multiples of 80, 85, and 90 separately. Out of all the multiples of 80, 85, and 90, we will focus on the multiples which are common to all numbers.

Then, out of all the common multiples, we will take out the smallest common multiple. That will be the LCM of 80, 85, and 90.

- Multiples of 80 are 80, 160, 240, 320, 400, 480, 560, 640, 720, 800,…, 12000, 12080, 12160, 12240, 12320, 12400,…
- Multiples of 85 are 85, 170, 255, 340, 425, 510, 595, 680, 765, 850,…, 11985, 12070, 12155, 12240, 12325, 12410,…
- Multiples of 90 are 90, 180, 270, 360, 450, 540, 630, 720, 810, 900,…, 11880, 11970, 12060, 12150, 12240, 12330, 12420,…

Here, it is clear that the least common multiple is 12240. So, the LCM of 80, 85, and 90 is 12240.

**Q5: If the product of three numbers is 612000, their HCF is 5, and the product of the HCF of the 1st and 2nd number and 2nd and 3rd number, and 1st and 3rd number among three numbers is 250. Find the LCM.Solution:**

As we know,

LCM of three numbers = (Product of three numbers * HCF) / [(HCF of 1st and 2nd number) * (HCF of 2nd and 3rd number) * (HCF 1st and 3rd number)]

It is given that,

product of the numbers = 612000, HCF = 5, and the product of the HCF of the 1st and 2nd number and 2nd and 3rd number, and 1st and 3rd number = 250

So, LCM = (612000 * 5) / 250

LCM = 3060000 / 250

LCM = 12240

Hence, the LCM is 12240.

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## Frequently Asked Questions (FAQs)

**What are the first three common numbers divisible by 80, 85, and 90?**

The first three common numbers divisible by 80, 85, and 90 are 12240, 24480, and 36720.

**Are the LCM of 80 and 85 the same as the LCM of 85 and 90?**

LCM of 80 and 85 is 1360 and LCM of 85 and 90 is 1530. So, the LCM of 80 and 85 are not the same as the LCM of 85 and 90.

**What is the LCM of 80, 85, and 90?**

LCM of 80, 85, and 90 is 12240.

**What are the methods to find the LCM of 80, 85, and 90?**

There are 3 major methods for finding the LCM of 80, 85, and 90:

1. Division Method

2. Listing Method

3. Prime Factorization Method

**Are LCM and HCF of 80, 85, and 90 the same?**

LCM of 80, 85, and 90 is 12240, and HCF of 80, 85, and 90 is 5. So, LCM and HCF of 80, 85, and 90 are not the same.

We hope you understand all the basics of how to find the LCM of 80, 85, and 90.

Written by

Prerit Jain