The smallest positive number that is a multiple of two or more numbers.
Let's start with an Example ...
List the Multiples of each number,
The multiples of 3 are 3, 6, 9, 12, 15, 18, ... etc
The multiples of 5 are 5, 10, 15, 20, 25, ... etc
Find the first Common (same) value:
The Least Common Multiple of 3 and 5 is 15
(15 is a multiple of both 3 and 5, and is the smallest number like that.)
So ... what is a "Multiple" ?
We get a multiple of a number when we multiply it by another number. Such as multiplying by 1, 2, 3, 4, 5, etc, but not zero. Just like the multiplication table.
Here are some examples:
| The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,... |
| The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,... |
What is a "Common Multiple" ?
Say we have listed the first few multiples of 4 and 5: the common multiples are those that are found in both lists:
| The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,... |
| The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,... |
Notice that 20 and 40 appear in both lists?
So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc ..., too)
What is the "Least Common Multiple" ?
It is simply the smallest of the common multiples.
In our previous example, the smallest of the common multiples is 20 ...
... so the Least Common Multiple of 4 and 5 is 20.
Finding the Least Common Multiple
List the multiples of the numbers until we get our first match.
The multiples of 4 are: 4, 8, 12, 16, 20, ...
and the multiples of 10 are: 10, 20, ...
Aha! there is a match at 20. It looks like this:
So the least common multiple of 4 and 10 is 20
The multiples of 6 are: 6, 12, 18, 24, 30, ...
and the multiples of 15 are: 15, 30, ...
There is a match at 30
So the least common multiple of 6 and 15 is 30
More than 2 Numbers
We can also find the least common multiple of three (or more) numbers.
Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Multiples of 6 are: 6, 12, 18, 24, 30, 36, ...
Multiples of 8 are: 8, 16, 24, 32, 40, ....
So 24 is the least common multiple (I can't find a smaller one!)
Hint: We can have smaller lists for the bigger numbers.
Least Common Multiple Tool
There is another method: the Least Common Multiple Tool does it automatically. (Yes, we waited until the end to tell you!)
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1 Expert Answer
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ... Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ... The LCM (Least Common Multiple) is the smallest number that appears on both lists.
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The LCM of 3 and 4 is 12.
Steps to find LCM
- Find the prime factorization of 3
3 = 3 - Find the prime factorization of 4
4 = 2 × 2 - Multiply each factor the greater number of times it occurs in steps i) or ii) above to find the LCM:
LCM = 2 × 2 × 3
- LCM = 12
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Find least common multiple (LCM) of: 6 & 8 9 & 12 15 & 20 21 & 28 6 & 4 3 & 8 9 & 4 3 & 12 15 & 4 3 & 20 21 & 4 3 & 28
Enter two numbers separate by comma. To find least common multiple (LCM) of more than two numbers, click here.
Least common multiple or lowest common denominator (lcd) can be calculated in two way; with the LCM formula calculation of greatest common factor (GCF), or multiplying the prime factors with the highest exponent factor.
Least common multiple (LCM) of 3 and 4 is 12.
LCM(3,4) = 12
Least Common Multiple of 3 and 4 with GCF Formula
The formula of LCM is LCM(a,b) = ( a × b) / GCF(a,b).
We need to calculate greatest common factor 3 and 4, than apply into the LCM equation.
GCF(3,4) = 1 LCM(3,4) = ( 3 × 4) / 1 LCM(3,4) = 12 / 1
LCM(3,4) = 12
Least Common Multiple (LCM) of 3 and 4 with Primes
Least common multiple can be found by multiplying the highest exponent prime factors of 3 and 4. First we will calculate the prime factors of 3 and 4.
Prime Factorization of 3
Prime factors of 3 are 3. Prime factorization of 3 in exponential form is:
3 = 31
Prime Factorization of 4
Prime factors of 4 are 2. Prime factorization of 4 in exponential form is:
4 = 22
Now multiplying the highest exponent prime factors to calculate the LCM of 3 and 4.
LCM(3,4) = 31 × 22
LCM(3,4) = 12
LCM of 3 and 4 is the smallest number among all common multiples of 3 and 4. The first few multiples of 3 and 4 are (3, 6, 9, 12, 15, 18, 21, . . . ) and (4, 8, 12, 16, 20, . . . ) respectively. There are 3 commonly used methods to find LCM of 3 and 4 - by listing multiples, by division method, and by prime factorization.
What is the LCM of 3 and 4?
Answer: LCM of 3 and 4 is 12.
Explanation:
The LCM of two non-zero integers, x(3) and y(4), is the smallest positive integer m(12) that is divisible by both x(3) and y(4) without any remainder.
Methods to Find LCM of 3 and 4
The methods to find the LCM of 3 and 4 are explained below.
- By Listing Multiples
- By Division Method
- By Prime Factorization Method
LCM of 3 and 4 by Listing Multiples
To calculate the LCM of 3 and 4 by listing out the common multiples, we can follow the given below steps:
- Step 1: List a few multiples of 3 (3, 6, 9, 12, 15, 18, 21, . . . ) and 4 (4, 8, 12, 16, 20, . . . . )
- Step 2: The common multiples from the multiples of 3 and 4 are 12, 24, . . .
- Step 3: The smallest common multiple of 3 and 4 is 12.
∴ The least common multiple of 3 and 4 = 12.
LCM of 3 and 4 by Division Method
To calculate the LCM of 3 and 4 by the division method, we will divide the numbers(3, 4) by their prime factors (preferably common). The product of these divisors gives the LCM of 3 and 4.
- Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 3 and 4. Write this prime number(2) on the left of the given numbers(3 and 4), separated as per the ladder arrangement.
- Step 2: If any of the given numbers (3, 4) 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 3 and 4 is the product of all prime numbers on the left, i.e. LCM(3, 4) by division method = 2 × 2 × 3 = 12.
LCM of 3 and 4 by Prime Factorization
Prime factorization of 3 and 4 is (3) = 31 and (2 × 2) = 22 respectively. LCM of 3 and 4 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 22 × 31 = 12.
Hence, the LCM of 3 and 4 by prime factorization is 12.
☛ Also Check:
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Example 1: Verify the relationship between GCF and LCM of 3 and 4.
Solution:
The relation between GCF and LCM of 3 and 4 is given as, LCM(3, 4) × GCF(3, 4) = Product of 3, 4
Prime factorization of 3 and 4 is given as, 3 = (3) = 31 and 4 = (2 × 2) = 22
LCM(3, 4) = 12 GCF(3, 4) = 1 LHS = LCM(3, 4) × GCF(3, 4) = 12 × 1 = 12 RHS = Product of 3, 4 = 3 × 4 = 12 ⇒ LHS = RHS = 12Hence, verified.
Example 2: The product of two numbers is 12. If their GCD is 1, what is their LCM?
Solution:
Given: GCD = 1 product of numbers = 12 ∵ LCM × GCD = product of numbers ⇒ LCM = Product/GCD = 12/1 Therefore, the LCM is 12.
The probable combination for the given case is LCM(3, 4) = 12.
Example 3: Find the smallest number that is divisible by 3 and 4 exactly.
Solution:
The smallest number that is divisible by 3 and 4 exactly is their LCM.
⇒ Multiples of 3 and 4:
- Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, . . . .
- Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, . . . .
Therefore, the LCM of 3 and 4 is 12.
Show Solution >
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The LCM of 3 and 4 is 12. To find the LCM of 3 and 4, we need to find the multiples of 3 and 4 (multiples of 3 = 3, 6, 9, 12; multiples of 4 = 4, 8, 12, 16) and choose the smallest multiple that is exactly divisible by 3 and 4, i.e., 12.
What is the Relation Between GCF and LCM of 3, 4?
The following equation can be used to express the relation between GCF and LCM of 3 and 4, i.e. GCF × LCM = 3 × 4.
If the LCM of 4 and 3 is 12, Find its GCF.
LCM(4, 3) × GCF(4, 3) = 4 × 3 Since the LCM of 4 and 3 = 12 ⇒ 12 × GCF(4, 3) = 12
Therefore, the greatest common factor (GCF) = 12/12 = 1.
Which of the following is the LCM of 3 and 4? 42, 28, 12, 36
The value of LCM of 3, 4 is the smallest common multiple of 3 and 4. The number satisfying the given condition is 12.
What are the Methods to Find LCM of 3 and 4?
The commonly used methods to find the LCM of 3 and 4 are:
- Listing Multiples
- Prime Factorization Method
- Division Method