LCM of 45 and 60

Find the Least Common Multiple (LCM) of 45 and 60. The LCM is the smallest number that both 45 and 60 divide evenly.

Answer: LCM(45, 60) = 180

Step-by-Step Solution

Step 1 List the Multiples of 45

To find the LCM of 45 and 60, we first list the multiples of each number.

Multiples of 45 are: 45 × 1, 45 × 2, 45 × 3, ...

Multiples of 45: 45, 90, 135, 180, 225, 270, ...

Step 2 List the Multiples of 60

Now we list the multiples of 60:

Multiples of 60: 60, 120, 180, 240, 300, ...

Step 3 Find the Least Common Multiple

The common multiples are numbers that appear in both lists.

The smallest common multiple is the LCM.

Looking at both lists, the first number that appears in both is 180.

LCM(45, 60) = 180

Step 4 Prime Factorization Method

We can also find the LCM using prime factorization:

45 = 3^2 × 5

60 = 2^2 × 3 × 5

Take each prime factor with the highest power:

LCM = 2^2 × 3^2 × 5 = 180

Step 5 Using the GCF Formula

The LCM can be calculated using the GCF with this formula:

LCM(a, b) = (a × b) ÷ GCF(a, b)

GCF(45, 60) = 15

LCM = (45 × 60) ÷ 15 = 2700 ÷ 15 = 180

Step 6 Final Answer

LCM(45, 60) = 180