What is the least common multiple of 2 and 3 and how to find it
When two numbers are multiplied together, the result is a multiple, or product. For instance, 20 is a multiple of 4 and 5 if we declare that 4 multiplied by 5 will give 20. The remaining multiples of four can be written as follows: 4, 8, 12, and so on as all of them are divisible by 4. Four times one equals four.
Common Multiples :
A whole number that is a shared multiple of every set of numbers is known as a common multiple. The term "common multiples" refers to the multiples that exist for two or more different numbers. On a grid of 100, let's mark the multiples of 6 and 7. The multiples of 6 will be marked with a circle, and the multiples of 7 with a cross.
Example of Common Multiples
Properties of Multiples :
The multiples' properties are listed below.
All numbers are multiples of one .
A number has an unlimited number of multiples.
A number's multiple is greater than or equal to the original number (except for 0).
How to Find Multiples of a Number:
When we multiply one whole number by another whole number, we have multiples as the result. Or, to put it simply, when you multiply it by a whole number, you receive its multiples.
Do you remember multiplication tables? They will be used by us to locate multiples. Let's identify the first five multiples of 6 that are not zero to see how that aids in our understanding of what multiples are.
Thus , Multiple of a number = Number Any integer (not a fraction).
Common Multiples of 2 and 3:
The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, ….
Hence, some of the common multiples of 2 and 3 are 6, 12, 18, etc.
Common Multiples of 3 and 5
The multiples of 3 are 3,6,9,12,15,18,21,24,27,30…..
The multiples of 5 are 5,10,15,20,25,30,35,40,45….
Hence, some common multiples of 3 and 5 are 15,30,45 etc
Conclusion :
A common multiple is a whole number that occurs multiple times in each set of numbers. A common multiple is a multiple that two or more numbers share.
Solved Examples:
1. List the first two multiples of 5 and 4 .
4, 8, 12, 16, 20, 24, 28, 32, 36, and 40,… are multiples of 4.
5, 10, 15, 20, 25, 30, 35, 40, and 45,... are multiples of 5.
20 and 40 are the two equivalent multiples of 5 and 4 respectively.
2. Find the least common multiples (LCM) of 3, 5 and 15.
First Let's list the multiples of 3, 5, and 15 that are common before determining the least common multiple.
Numerous multiples of 3 include 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and so forth.
5, 10, 15, 20, 25, 30, 35, 40, 45, and so on are examples of multiples of 5.
15, 30, 45, 60, 75, 90, 105, and other multiples of 15
15, 30,... and so on are common multiples of 3, 5, and 15 respectively. The smallest multiple among these regular multiples is 15. Therefore, the LCM of 3, 5, and 15 is 15 because it is the least of all the common multiples. Thus, 15 is the LCM of 3, 5 and 15.
3. List the first three multiples of 2 and 6 that are frequently repeated.
Numerous 2's include 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and 24.
Numerous 6 include: 6, 12, 18, 24, 30, 36, 42, and 48.
The first two common multiples of 2 and 6 are 6 and 12.
FAQs on Common Multiple of 2 and 3 with Definition and Examples
1. What is the common multiple of 2 and 3?
The smallest common multiple of 2 and 3 is 6. A common multiple is a number that is divisible by both numbers exactly.
- Multiples of 2: 2, 4, 6, 8, 10, 12, …
- Multiples of 3: 3, 6, 9, 12, 15, …
- The common multiples are: 6, 12, 18, 24, …
2. What is the LCM of 2 and 3?
The LCM of 2 and 3 is 6. The Least Common Multiple is the smallest positive number that both numbers divide exactly.
- Prime factors of 2 = 2
- Prime factors of 3 = 3
- LCM = 2 × 3 = 6
3. How do you find the common multiples of 2 and 3?
You find the common multiples of 2 and 3 by listing their multiples or by using the LCM method. The key result is that all common multiples are multiples of 6.
- List multiples of 2 and 3 separately.
- Identify numbers that appear in both lists.
- Or calculate LCM(2,3) = 6 and write 6, 12, 18, 24, …
4. Why is 6 the least common multiple of 2 and 3?
The number 6 is the least common multiple of 2 and 3 because it is the smallest number divisible by both without remainder.
- 6 ÷ 2 = 3 (no remainder)
- 6 ÷ 3 = 2 (no remainder)
5. What are the first five common multiples of 2 and 3?
The first five common multiples of 2 and 3 are 6, 12, 18, 24, and 30. These are found by multiplying the LCM (6) by natural numbers.
- 6 × 1 = 6
- 6 × 2 = 12
- 6 × 3 = 18
- 6 × 4 = 24
- 6 × 5 = 30
6. Are there infinitely many common multiples of 2 and 3?
Yes, there are infinitely many common multiples of 2 and 3. Any multiple of their LCM (6) is a common multiple.
- 6, 12, 18, 24, 30, …
- General form: 6 × n, where n is a positive integer
7. What is the formula to find the LCM of 2 and 3 using GCD?
The formula to find LCM using GCD is LCM(a, b) = (a × b) ÷ GCD(a, b). For 2 and 3:
- GCD(2,3) = 1
- LCM = (2 × 3) ÷ 1
- LCM = 6
8. What is the difference between a multiple and a common multiple?
A multiple is the result of multiplying a number by an integer, while a common multiple is a number that is a multiple of two or more numbers.
- Multiples of 2: 2, 4, 6, 8, …
- Multiples of 3: 3, 6, 9, …
- Common multiples of 2 and 3: 6, 12, 18, …
9. Can you give a real-life example of common multiples of 2 and 3?
A real-life example of common multiples of 2 and 3 is scheduling events that repeat every 2 and 3 days, which will coincide every 6 days.
- Event A happens every 2 days.
- Event B happens every 3 days.
- Both occur together on day 6, 12, 18, …
10. Is every multiple of 6 a common multiple of 2 and 3?
Yes, every multiple of 6 is a common multiple of 2 and 3. This is because 6 is divisible by both numbers.
- 6 ÷ 2 = 3 and 6 ÷ 3 = 2
- 12 ÷ 2 = 6 and 12 ÷ 3 = 4
- 18 ÷ 2 = 9 and 18 ÷ 3 = 6
