Question

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 1

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.