LCM of 6 and 10

LCM of 6 and 10 is 30. LCM or Least Common Multiple is the smallest multiple of both 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 6.

What is LCM of 6 and 10?

Answer: LCM of 6 and 10 is 30.

Various techniques, including prime factorization, listing multiples, or the ladder method, can be utilized to find the LCM. For instance, the prime factorization of 6 is represented as 2 × 3, and for 10 it is 2 × 5. Through the multiplication of the highest powers of these prime factors, the LCM of 6 and 10 is determined as 2 × 3 × 5, resulting in a value of 30.

Thus, it can be affirmed that the LCM of 6 and 10 is 30.

Before moving further we must first learn in brief about LCM.

What is LCM?

Least Common Multiple, or LCM, is the smallest positive integer that is divisible by each of the given numbers without leaving a remainder. It is a fundamental concept in arithmetic that finds extensive use in various mathematical problems.

LCM of 6 and 10 Calculator

How to Find LCM of 6 and 10

To find LCM of 6 and 10, we use three different methods that include,

• LCM of 6 and 10 by Prime Factorization
• LCM of 6 and 10 by Listing Multiple
• LCM of 6 and 10 by Long Division

LCM of 6 and 10 by the method of Prime Factorization

Prime factorization involves breaking down each number into its prime factors and identifying the common factors to calculate the LCM efficiently.

• Prime Factorization of 6: 2×3
• Prime Factorization of 10: 2×5

LCM is obtained by multiplying the highest powers of prime factors: 2×3×5 = 30

Therefore, LCM of 6 and 10 by prime factorization is 30.

Learn more about, Prime Factorization Method

LCM of 6 and 10 by the method of Listing Multiples

Listing multiples is another approach to finding the LCM. Identify the multiples of each number and determine the smallest value they have in common.

• Multiples of 6 are: 6, 12, 18, 24, 30…
• Multiples of 10 are: 10, 20, 30, 40…

Here, 30 is the first common multiple of both 6 and 10.

Therefore, LCM of 6 and 10 by listing multiples is 30.

LCM of 6 and 10 by the method of Long Division

Long division is a systematic method for finding the LCM. Divide multiples of the larger number until a common multiple is reached, providing the LCM. We divide the numbers 6 and 10 by their prime factors to determine their LCM. The product of these divisors shows the least common multiple of 6 and 10 as shown below:

LCM and HCF of 6 and 10

LCM represents the smallest common multiple, the Highest Common Factor (HCF) is the greatest number that divides both 10 and 6. Understanding both concepts is significant for comprehensive mathematical applications.

Also, the important property of LCM and HCF of 6 and 10 is,

LCM of 6 and 10 × HCF of 6 and 10 = Product of 6 and 10

30 × 2 = 6 × 10

60 = 60

Solved Examples on LCM of 6 and 10

Some examples on LCM of 6 and 10

Example 1: You’re planning a party and want to serve both cupcakes (6 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:

LCM of 6 and 10 is 30

Therefore, you need to bake at least 5 batches of cupcakes (6 × 5 = 30) and 3 batches of cookies (10 × 3 = 30) to have 30 cupcakes and 30 cookies, enough for everyone to enjoy!

Example 2: Two friends, Maya and Alex, have different workout routines. Maya trains after every 6th day, while Alex does yoga after every 10th day. They starts at 1 of Jan and want to find the next time their workout schedules will coincide.

Solution:

LCM of 6 and 10 is 30

So, Maya and Alex will both be working out on the same day again every 30 days, the “common beat” of their schedules.

Example 3: Find the LCM of 10 and 6 using the listing multiples method.

Solution:

Multiples of 10: 10, 20, 30, 40, 50…

Multiples of 6: 6, 12, 18, 24, 30, 36…

Common multiple: 30

LCM(10, 6) = 30

Example 4: Explore the factors of 10 and 6 by choosing a larger set of prime factors and then using prime factorization calculate its LCM.

Solution:

10 = 2×5

6 = 2×3

Common Factor: 2

Uncommon Factors: 5 and 3

LCM(10,6) = 2×5×3 =30

Practice Problems on LCM of 6 and 10

Some practice problems on LCM of 6 and 10 are,

P1: Find the LCM of 6 and 10 using the prime factorization method?

P2: List the first three common multiples of 6 and 10?

P3: Calculate the LCM of 6 and 10 using the listing multiples method?

P4: If the LCM of 6 and 10 is 30, what are the possible pairs of numbers that satisfy this LCM?

P5: Determine the LCM of 6 and 10 using the ladder method?

FAQS on LCM of 6 and 10

What is Least Common Multiple (LCM) of 6 and 10?

LCM of 6 and 10 is 30.

How is LCM calculated for 6 and 10?

To find the LCM, methods like prime factorization, listing multiples, or the ladder method can be used.

Can LCM of 6 and 10 be Smaller than both Numbers?

No, LCM is always equal to or greater than the given numbers. For 6 and 10, the LCM 30 is greater than both 6 and 10.

Why is finding the LCM important?

Finding the LCM is crucial in various mathematical operations, such as simplifying fractions, solving equations, and working with common denominators.

What are Common Multiples of 6 and 10?

Common multiples of 6 and 10 are multiples of their LCM (30). Examples 30, 60, 90, and so on.

Can LCM of 6 and 10 be Negative?

No, LCM is always a positive integer. It represents the smallest positive multiple that is divisible by both numbers.

Is Order of Numbers Important When Finding LCM?

No, the LCM is the same regardless of the order of the numbers. The LCM of 6 and 10 is the same as the LCM of 10 and 6, which is 30.