Question

How do find the LCM of 27 and 45 by listing multiples?

Answer 1

Hint: We write the prime factorization of each of the given numbers and then calculate the least common multiple using their prime factors.
* LCM is the least common multiple of two or more numbers. We write each number in the form of its prime factors and write the LCM of the numbers as multiplication of prime factors with their highest powers.
* Prime factorization is a process of writing a number in multiple of its factors where all factors are prime numbers.
* Law of exponent states that when the base is same we can add the powers of that element i.e. \[{p^m} \times {p^n} = {p^{m + n}}\]

Complete step-by-step answer:
We have to calculate the LCM of the numbers 27 and 45.
We will write the prime factorization of the given numbers i.e. we write the numbers as the product of their prime factors.
We can write
\[27 = 3 \times 3 \times 3\]
\[54 = 2 \times 3 \times 3 \times 3\]
We can collect the powers of numbers having same base using the rule of exponents
\[27 = {3^3}\]
\[54 = 2 \times {3^3}\]
From the prime factorization of the two numbers we see that the highest power of prime number 2 is 1 and 3 is 3.
\[ \Rightarrow \]LCM \[ = 2 \times 3 \times 3 \times 3\]
\[ \Rightarrow \]LCM \[ = 54\]

\[\therefore \]LCM of the two numbers 27 and 54 is 54.

Note:
Students might get confused when calculating the LCM as they only take the common factors between the prime factorizations of the two numbers as LCM is least common multiple and they only take common factors which are wrong. Keep in mind we have to calculate the smallest number which is multiple of both the given numbers.