LCM of 8 and 10

LCM of 8 and 10 is 40. LCM stands for Least Common Multiple. LCM is used to find the smallest common multiple of two or more numbers. The concept of LCM is very important and is used in various problems of mathematics.

In this article, we will discuss the concept of LCM and specifically explore its calculation for the numbers 10 and 8 by different methods.

What is LCM?

Least Common Multiple (LCM) of two or more numbers is the smallest number that is divisible by all of them. It is often denoted by LCM(a, b) for two numbers a and b. In the case of 8 and 10, it is the smallest number that can be divided by both 8 and 10 without leaving any remainder.

What is LCM of 8 and 10?

LCM of 8 and 10 is 40.

Various techniques including prime factorization, listing multiples, or the long division method, can be utilized to find the LCM. For instance, the prime factorization of 8 is represented as 2 × 2 × 2, and for 10 it is 2 × 5. By multiplying the highest powers of these prime factors, the LCM of 8 and 10 is determined as 2 × 2 × 2 × 5, resulting in a value of 40.

Thus, it can be affirmed that the LCM of 8 and 10 is 40.

LCM of 8 and 10 Calculator

How to Find LCM of 8 and 10

To find LCM of 8 and 10, we use three different methods that include,

• Prime Factorization
• Listing Multiple
• Division Method

LCM of 8 and 10 by Prime Factorization

To find the LCM of 8 and 10 using prime factorization, we first write each number into its prime factors. For 8, it is 2³ and for 10, it 2 × 5. Then, we take the highest power of each prime factor present in either number.

Below is the step-by-step illustration to find LCM of 8 and 10 by prime factorization method:

Step 1: Find the Prime Factorization of 8 and 10

• Prime Factorization of 8: 2 × 2 × 2

• Prime Factorization of 10: 2 × 5

Step 2: Identify the Highest Powers of prime factors of 8 and 10

• The highest power of 2 is 2³

• The highest power of 5 is 5¹

Step 3: Multiply the Highest Powers

LCM of 8 and 10 is 2³ × 5 = 40

LCM is obtained by multiplying the highest powers of prime factors: 2 × 2 × 2 × 5 = 40

Therefore, LCM of 8 and 10 by prime factorization is 40.

LCM of 8 and 10 by Listing Multiples

To find the LCM, we can use method of listing the multiples of each number until we find a common multiple. Below is the step-by-step illustration to find LCM of 8 and 10 by listing multiples:

Step 1: List the Multiples of 8 and 10

• The multiples of 8 are 8, 16, 24, 32, 40, 48……
• The multiples of 10 are 10, 20, 30, 40, 50 ……..

Step 2: Identify the smallest common multiple

Here, we see that 40 is the smallest common multiple of 8 and 10.

Therefore, LCM of 8 and 10 is 40.

LCM of 8 and 10 by Division Method

We can also use the division method to find the LCM of two or more numbers. Long division is a systematic method for finding the LCM. Steps involved in Division method are given below:

• Divide the numbers 8 and 10 by smallest prime number to determine their LCM.

• If any number is not divisible, then write down that number and proceed further.

• Keep on dividing the row of numbers by prime numbers, unless we get the results as 1 in the complete row.

• Now LCM of the numbers will be equal to the product of all the prime numbers we obtained in the division method.

To calculate the LCM of 8 and 10 by division method, we will divide the numbers by their prime factors as shown:

LCM and HCF of 8 and 10

Highest common factor (HCF) of 8 and 10 is 2, and the least common multiple (LCM) is 40. HCF represents the largest number that divides both 8 and 10 without leaving a remainder, while the LCM represents the smallest number divisible by both 8 and 10.

LCM and HCF of 8 and 10 also follow product rule where product of two numbers is equal to the product of their respective LCM and HCF. The mathematical representation of this formula is given below:

LCM of 8 and 10 × HCF of 8 and 10 = Product of 8 and 10

40 × 2 = 8 × 10

80 = 80

Hence proved.

Solved Examples on LCM of 8 and 10

Example 1: You’re planning a party and want to serve both cupcakes (8 per batch) and cookies (10 per batch) to your guests. To ensure everyone gets an equal share, you need to find the smallest number of batches for each treat that results in the same number of cupcakes and cookies.

Solution:

Smallest number of batches for each treat that results in the same number of cupcakes and cookies is the LCM the number of treats in each batch.

LCM of 8 and 10 is 40.

Therefore, you need to bake at least 5 batches of cupcakes (8 × 5 = 40) and 4 batches of cookies (10 × 4 = 40) to have 40 cupcakes and 40 cookies.

Example 2: Given the factors of 10 and 8 using prime factorization and calculate their LCM.

Solution:

Prime factors of 10 = 2 × 5

Prime factors of 8 = 2 × 2 ×2

LCM is obtained by multiplying the highest powers of prime factors: 2 × 2 × 2 × 5 = 40

Hence, LCM (10,8) = 2 × 5 × 2 × 2 =40

Example 3: Two bells ring after a fixed interval. First bell rings after every 8 hours, while second bell rings after every 10 hours. Starting now, when the next time will they ring together?

Solution:

To find the time when the two bells will ring together, we have to find the LCM of time interval of their occurrence i.e. 8 and 10

LCM of 8 and 10 is 40

Hence, two bells will ring together after every 40 hours

Example 4: Find the LCM of 10 and 8 using the listing multiples method.

Solution:

Multiples of 10: 10, 20, 30, 40, 50…

Multiples of 8: 8, 16, 24, 32, 40, 48…

Common multiple: 40

LCM (10, 8) = 40

Practice Problems on LCM of 8 and 10

Problem 1: Find the LCM of 8 and 10 using the prime factorization method?

Problem 2: List the first three common multiples of 8 and 10?

Problem 3: Calculate the LCM of 8 and 10 using the listing multiples method?

Problem 4: Determine the LCM of 8 and 10, given HCF(8, 10) = 2?

Problem 5: If two events occur at intervals of 10 and 8 hours, when will they next occur simultaneously?

LCM of 8 and 10: FAQs

What is Least Common Multiple (LCM) of 8 and 10?

LCM of 8 and 10 is 40.

What are Methods to Calculate LCM of 8 and 10?

Methods like prime factorization, listing multiples or the ladder method are adopted for LCM calculation for 8 and 10

Can LCM be Smaller than the given Numbers?

No, LCM is always equal to or greater than the given numbers.

Can LCM of 8 and 10 be Negative?

LCM is always a positive integer. It cannot be negative.

What is the LCM of 10, 8, and 9?

LCM of 10, 8 and 9 is 360.