least common multiple 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 the LCM, we can use the prime factorization method or find the multiples of both numbers until we identify the smallest common one.

Answer

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

Steps to Solve

Find Prime Factorization of Each Number

Start by breaking down each number into its prime factors.

For $60$, the prime factorization is: $$ 60 = 2^2 \times 3^1 \times 5^1 $$

For $90$, the prime factorization is: $$ 90 = 2^1 \times 3^2 \times 5^1 $$

Identify the Highest Powers of Each Prime Factor

Next, for each prime factor, identify the highest power that appears in either factorization.

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

Multiply the Highest Powers Together

Now, multiply these highest powers together to find the least common multiple (LCM):

$$ \text{LCM} = 2^2 \times 3^2 \times 5^1 $$

Calculating this:

$2^2 = 4$
$3^2 = 9$
$5^1 = 5$

Now, multiply these values: $$ \text{LCM} = 4 \times 9 \times 5 $$

Calculate the Final LCM

First, compute $4 \times 9$: $$ 4 \times 9 = 36 $$

Then multiply by 5: $$ 36 \times 5 = 180 $$

Thus, the least common multiple of 60 and 90 is $180$.

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

More Information

The least common multiple (LCM) represents the smallest number that both original numbers can divide into without leaving a remainder. Understanding the LCM is useful in various applications, such as scheduling and problem-solving in fractions.

Tips

Forgetting to take the highest power of each prime factor.
Mixing up the multiplication order or failing to multiply all identified factors.
Not checking if you've included all prime factors from both numbers.

Thank you for voting!