What is the least common multiple of 9 and 18

What is the Least Common Multiple (LCM) of 4, 9, and 18? Here we will show you step-by-step how to find the Least Common Multiple of 4, 9, and 18.

Step 1) First we find and list the prime factors of 4, 9, and 18 (Prime Factorization):

Prime Factors of 4:
2, 2

Prime Factors of 9:
3, 3

Prime Factors of 18:
2, 3, 3

2, 2, 3, 3 Step 3) Finally, we multiply the prime numbers from Step 2 together.

2 x 2 x 3 x 3 = 36 That's it. The Least Common Multiple (LCM) of 4, 9, and 18 is 36.

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Least Common Multiple (LCM) of 4, 9, and 19
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Least common multiple can be found by multiplying the highest exponent prime factors of 9 and 18. First we will calculate the prime factors of 9 and 18.

Prime Factorization of 9

Prime factors of 9 are 3. Prime factorization of 9 in exponential form is:

9 = 32

Prime Factorization of 18

Prime factors of 18 are 2,3. Prime factorization of 18 in exponential form is:

18 = 21×32

Now multiplying the highest exponent prime factors to calculate the LCM of 9 and 18.

LCM(9,18) = 21×32
LCM(9,18) = 18

Factors of 9

List of positive integer factors of 9 that divides 9 without a remainder.

1, 3, 9

Factors of 18

List of positive integer factors of 18 that divides 18 without a remainder.

1, 2, 3, 6, 9, 18

The formula of LCM is LCM(a,b) = ( a × b) / GCF(a,b).
We need to calculate greatest common factor 9 and 18, than apply into the LCM equation.

GCF(9,18) = 9
LCM(9,18) = ( 9 × 18) / 9
LCM(9,18) = 162 / 9
LCM(9,18) = 18

(i) The LCM of 18 and 9 is associative

LCM of 9 and 18 = LCM of 18 and 9

1. What is the LCM of 9 and 18?

Answer: LCM of 9 and 18 is 18.

2. What are the Factors of 9?

Answer: Factors of 9 are 1, 3, 9. There are 3 integers that are factors of 9. The greatest factor of 9 is 9.

3. What are the Factors of 18?

Answer: Factors of 18 are 1, 2, 3, 6, 9, 18. There are 6 integers that are factors of 18. The greatest factor of 18 is 18.

4. How to Find the LCM of 9 and 18?

Answer:

Least Common Multiple of 9 and 18 = 18

Step 1: Find the prime factorization of 9

9 = 3 x 3

Step 2: Find the prime factorization of 18

18 = 2 x 3 x 3

Step 3: Multiply each factor the greater number of times it occurs in steps i) or ii) above to find the lcm:

LCM = 18 = 2 x 3 x 3

Step 4: Therefore, the least common multiple of 9 and 18 is 18.

Explanation:

list the multiples of each number and pick out the lowest common one

multiples of #9:" "{9,color(red)(18),27,color(red)(36),45,color(red)(54),...}#

multiples of #18:" "{color(red)(18,36,54...)}#

common multiples#" "{18,36,54,...}#

#lcm(9,18)=18#

Answer

Verified

Hint: Here we will use the prime factorization method to find the L.C.M. First we will write the given numbers as the product of their prime factors one – by – one. Now, if a prime factor will be repeating then we will write them in exponential form. Finally, we will take the product of all the different prime factors along with their highest exponent to get the answer.

Complete step-by-step solution:
Here we have been asked to find the least common multiple of the numbers: 9, 18 and 21. First, let us know about L.C.M.
In arithmetic and number theory, the least common multiple (L.C.M) of two or more integers is the smallest positive integer that is divisible by each of the given numbers. The given integers must not be 0. There are two methods to determine the L.C.M of two or more given numbers. Here, we will use the method of prime factorization.
In the method of prime factorization we write the given numbers as the product of their prime factors. Now, the L.C.M will be the product of the prime factors along with their highest exponent present.
Now let us come to the question. Here we have three numbers: 9, 18 and 21. Using prime factorization we get,
\[\begin{align}
  & \Rightarrow 9=3\times 3={{3}^{2}} \\
 & \Rightarrow 18=2\times 3\times 3=2\times {{3}^{2}} \\
 & \Rightarrow 21=3\times 7 \\
\end{align}\]
Clearly we can see that the highest power of the prime factors 2 is 1, 3 is 2 and 7 is 1. So, we need to multiply ${{2}^{1}}$, ${{3}^{2}}$ and ${{7}^{1}}$ to get the L.C.M.
\[\Rightarrow \] L.C.M = \[{{2}^{1}}\times {{3}^{2}}\times {{7}^{1}}\]
\[\Rightarrow \] L.C.M = \[2\times 9\times 7\]
$\therefore $ L.C.M = 126
Hence, the L.C.M of 9, 18 and 18 is 126.

Note: There is one more method by which we can find the L.C.M. In that method we will write the multiples of 9, 18 and 21 one – by – one and check which multiple occurs first. But this method may not be preferred more because initially we don’t know how many multiples we need to write. In case the numbers provided are large then prime factorization is the best approach. Do not get confused with the process of determining the L.C.M with that of determining the H.C.F. Remember that while finding the H.C.F we only consider the factors which are common in all the given numbers.

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