LCM of 60 and 90

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 60 and 90. To find this, we can use the prime factorization method or the method of listing multiples.

Answer

The least common multiple of 60 and 90 is $ 180 $.

Steps to Solve

Find Prime Factorization of Each Number

First, let's factor the numbers 60 and 90 into their prime factors.

For 60, we can break it down as: $$ 60 = 2^2 \times 3^1 \times 5^1 $$

For 90, we can break it down as: $$ 90 = 2^1 \times 3^2 \times 5^1 $$

Identify the Highest Powers of Each Prime Factor

Next, we identify the highest power of each prime that appears in the factorizations of 60 and 90.

For the prime number 2, the highest power is $2^2$ (from 60).
For the prime number 3, the highest power is $3^2$ (from 90).
For the prime number 5, the highest power is $5^1$ (both have $5^1$).

Calculate the LCM Using the Prime Factors

Now we multiply these highest powers together to find the LCM.

So, $$ \text{LCM}(60, 90) = 2^2 \times 3^2 \times 5^1 $$

Calculating this gives: $$ \text{LCM}(60, 90) = 4 \times 9 \times 5 $$

Final Calculation for LCM

Finally, we can calculate this step-by-step:

First calculate $4 \times 9 = 36$.
Then calculate $36 \times 5 = 180$.

So, $$ \text{LCM}(60, 90) = 180 $$

The least common multiple of 60 and 90 is ( \text{LCM}(60, 90) = 180 ).

More Information

The least common multiple (LCM) is the smallest multiple that is exactly divisible by each of the given numbers. It is useful in various applications, such as finding a common denominator in fractions.

Tips

Incorrect Prime Factorization: Ensure that each number is factored correctly to avoid calculation errors. Double-check the multiplication of factors.
Not Using Highest Powers: Remember to select the highest power for each prime factor when determining the LCM.

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