What is the least common multiple of 6 and 18

Least common multiple can be found by multiplying the highest exponent prime factors of 6 and 18. First we will calculate the prime factors of 6 and 18.

Prime Factorization of 6

Prime factors of 6 are 2,3. Prime factorization of 6 in exponential form is:

6 = 21×31

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 6 and 18.

LCM(6,18) = 21×32
LCM(6,18) = 18

Factors of 6

List of positive integer factors of 6 that divides 6 without a remainder.

1, 2, 3, 6

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 6 and 18, than apply into the LCM equation.

GCF(6,18) = 6
LCM(6,18) = ( 6 × 18) / 6
LCM(6,18) = 108 / 6
LCM(6,18) = 18

(i) The LCM of 18 and 6 is associative

LCM of 6 and 18 = LCM of 18 and 6

1. What is the LCM of 6 and 18?

Answer: LCM of 6 and 18 is 18.

2. What are the Factors of 6?

Answer: Factors of 6 are 1, 2, 3, 6. There are 4 integers that are factors of 6. The greatest factor of 6 is 6.

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 6 and 18?

Answer:

Least Common Multiple of 6 and 18 = 18

Step 1: Find the prime factorization of 6

6 = 2 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 6 and 18 is 18.

Three common multiples of 18 and 6 are(a) 18, 6, 9(b) 18, 36, 6(c) 36, 54, 72(d) None of these

Answer

Verified

Hint: To calculate three common multiples of 18 and 6, calculate the LCM of the given two numbers. Find other multiples of the LCM by multiplying it by and positive integers. Those multiples of LCM along with the LCM will be common multiples of the given numbers.

Step-by-step answer:
We have to calculate the common multiples of 18 and 6. To do so, we will first evaluate the LCM 18 and 6.
To find the LCM of the given numbers, we will use the prime factorization method. We will write the prime factorization of each of the numbers in exponential form. Then, we will align the common prime factor base whenever possible. For the numbers with a common prime factor base, select the prime number that has the highest power. The prime factor with the highest power implies that it occurs the most in the list. If a distinct prime factor has no matching prime factor base in the list, include this factor with its exponent in the collection of numbers. Multiply all the numbers which were collected earlier to get the LCM of the numbers.
The prime factorization of 18 is $18=2\times {{3}^{2}}$. The prime factorization of 6 is $6=2\times 3$.
Thus, the LCM of 18 and 6 is $2\times {{3}^{2}}=18$.
We will now find other common multiples of 18 and 6. To do so, we will multiply the LCM by 2, 3, and 4.
Thus, we have $18\times 2=36,18\times 3=54,18\times 4=72$.
Hence, the common multiples of 18 and 6 are 36, 54, and 72, which is option (c).

Note: We must keep in mind that the least common multiple (LCM) of 18 and 6 is 18. This is because 18 is a multiple of 6. We can obtain other common multiples of 18 and 6 by multiplying the LCM of 18 and 6 by any positive integer.

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 6 and 18 is 18.

LCM(6,18) = 18

Least Common Multiple of 6 and 18 with GCF Formula

The formula of LCM is LCM(a,b) = ( a × b) / GCF(a,b).
We need to calculate greatest common factor 6 and 18, than apply into the LCM equation.

GCF(6,18) = 6
LCM(6,18) = ( 6 × 18) / 6
LCM(6,18) = 108 / 6
LCM(6,18) = 18

Least Common Multiple (LCM) of 6 and 18 with Primes

Least common multiple can be found by multiplying the highest exponent prime factors of 6 and 18. First we will calculate the prime factors of 6 and 18.

Prime Factorization of 6

Prime factors of 6 are 2, 3. Prime factorization of 6 in exponential form is:

6 = 21 × 31

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 6 and 18.

LCM(6,18) = 21 × 32
LCM(6,18) = 18

What is the LCM and HCF of 6 and 18?

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