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LCM of 80, 85, and 90

LCM of 80, 85, and 90 is the smallest number among all common multiples of 80, 85, and 90. The first few multiples of 80, 85, and 90 are (80, 160, 240, 320, 400 . . .), (85, 170, 255, 340, 425 . . .), and (90, 180, 270, 360, 450 . . .) respectively. There are 3 commonly used methods to find LCM of 80, 85, 90 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 80, 85, and 90
2.	List of Methods
3.	Solved Examples
4.	FAQs

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

Answer: LCM of 80, 85, and 90 is 12240.

Explanation:

The LCM of three non-zero integers, a(80), b(85), and c(90), is the smallest positive integer m(12240) that is divisible by a(80), b(85), and c(90) without any remainder.

Methods to Find LCM of 80, 85, and 90

Let's look at the different methods for finding the LCM of 80, 85, and 90.

By Prime Factorization Method
By Division Method
By Listing Multiples

LCM of 80, 85, and 90 by Prime Factorization

Prime factorization of 80, 85, and 90 is (2 × 2 × 2 × 2 × 5) = 2⁴ × 5¹, (5 × 17) = 5¹ × 17¹, and (2 × 3 × 3 × 5) = 2¹ × 3² × 5¹ respectively. LCM of 80, 85, and 90 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁴ × 3² × 5¹ × 17¹ = 12240.
Hence, the LCM of 80, 85, and 90 by prime factorization is 12240.

LCM of 80, 85, and 90 by Division Method

To calculate the LCM of 80, 85, and 90 by the division method, we will divide the numbers(80, 85, 90) by their prime factors (preferably common). The product of these divisors gives the LCM of 80, 85, and 90.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 80, 85, and 90. Write this prime number(2) on the left of the given numbers(80, 85, and 90), separated as per the ladder arrangement.
Step 2: If any of the given numbers (80, 85, 90) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 80, 85, and 90 is the product of all prime numbers on the left, i.e. LCM(80, 85, 90) by division method = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 17 = 12240.

LCM of 80, 85, and 90 by Listing Multiples

To calculate the LCM of 80, 85, 90 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 80 (80, 160, 240, 320, 400 . . .), 85 (85, 170, 255, 340, 425 . . .), and 90 (90, 180, 270, 360, 450 . . .).
Step 2: The common multiples from the multiples of 80, 85, and 90 are 12240, 24480, . . .
Step 3: The smallest common multiple of 80, 85, and 90 is 12240.

∴ The least common multiple of 80, 85, and 90 = 12240.

☛ Also Check:

LCM of 80, 85, and 90 Examples

Example 1: Verify the relationship between the GCD and LCM of 80, 85, and 90.

Solution:

The relation between GCD and LCM of 80, 85, and 90 is given as,
LCM(80, 85, 90) = [(80 × 85 × 90) × GCD(80, 85, 90)]/[GCD(80, 85) × GCD(85, 90) × GCD(80, 90)]
⇒ Prime factorization of 80, 85 and 90:
- 80 = 2⁴ × 5¹
- 85 = 5¹ × 17¹
- 90 = 2¹ × 3² × 5¹
∴ GCD of (80, 85), (85, 90), (80, 90) and (80, 85, 90) = 5, 5, 10 and 5 respectively.
Now, LHS = LCM(80, 85, 90) = 12240.
And, RHS = [(80 × 85 × 90) × GCD(80, 85, 90)]/[GCD(80, 85) × GCD(85, 90) × GCD(80, 90)] = [(612000) × 5]/[5 × 5 × 10] = 12240
LHS = RHS = 12240.
Hence verified.
Example 2: Find the smallest number that is divisible by 80, 85, 90 exactly.

Solution:

The value of LCM(80, 85, 90) will be the smallest number that is exactly divisible by 80, 85, and 90.
⇒ Multiples of 80, 85, and 90:
- Multiples of 80 = 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, . . . ., 12080, 12160, 12240, . . . .
- Multiples of 85 = 85, 170, 255, 340, 425, 510, 595, 680, 765, 850, . . . ., 12070, 12155, 12240, . . . .
- Multiples of 90 = 90, 180, 270, 360, 450, 540, 630, 720, 810, 900, . . . ., 11970, 12060, 12150, 12240, . . . .
Therefore, the LCM of 80, 85, and 90 is 12240.
Example 3: Calculate the LCM of 80, 85, and 90 using the GCD of the given numbers.

Solution:

Prime factorization of 80, 85, 90:
- 80 = 2⁴ × 5¹
- 85 = 5¹ × 17¹
- 90 = 2¹ × 3² × 5¹
Therefore, GCD(80, 85) = 5, GCD(85, 90) = 5, GCD(80, 90) = 10, GCD(80, 85, 90) = 5
We know,
LCM(80, 85, 90) = [(80 × 85 × 90) × GCD(80, 85, 90)]/[GCD(80, 85) × GCD(85, 90) × GCD(80, 90)]
LCM(80, 85, 90) = (612000 × 5)/(5 × 5 × 10) = 12240
⇒LCM(80, 85, 90) = 12240

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FAQs on LCM of 80, 85, and 90

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

The LCM of 80, 85, and 90 is 12240. To find the LCM (least common multiple) of 80, 85, and 90, we need to find the multiples of 80, 85, and 90 (multiples of 80 = 80, 160, 240, 320 . . . . 12240 . . . . ; multiples of 85 = 85, 170, 255, 340 . . . . 12240 . . . . ; multiples of 90 = 90, 180, 270, 360 . . . . 12240 . . . . ) and choose the smallest multiple that is exactly divisible by 80, 85, and 90, i.e., 12240.

What are the Methods to Find LCM of 80, 85, 90?

The commonly used methods to find the LCM of 80, 85, 90 are:

Prime Factorization Method
Division Method
Listing Multiples

What is the Relation Between GCF and LCM of 80, 85, 90?

The following equation can be used to express the relation between GCF and LCM of 80, 85, 90, i.e. LCM(80, 85, 90) = [(80 × 85 × 90) × GCF(80, 85, 90)]/[GCF(80, 85) × GCF(85, 90) × GCF(80, 90)].

Which of the following is the LCM of 80, 85, and 90? 12240, 3, 10, 36

The value of LCM of 80, 85, 90 is the smallest common multiple of 80, 85, and 90. The number satisfying the given condition is 12240.

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