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 solve this, we will find the prime factorization of both numbers and then determine the LCM based on their factors.

Answer

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

Steps to Solve

Find the prime factorization of 60

To find the prime factorization of 60, we divide by the smallest prime numbers until we reach 1.

$$ 60 = 2 \times 30 $$ $$ 30 = 2 \times 15 $$ $$ 15 = 3 \times 5 $$

So, the prime factorization of 60 is:

$$ 60 = 2^2 \times 3^1 \times 5^1 $$

Find the prime factorization of 90

Now let's find the prime factorization of 90 in the same way.

$$ 90 = 2 \times 45 $$ $$ 45 = 3 \times 15 $$ $$ 15 = 3 \times 5 $$

So, the prime factorization of 90 is:

$$ 90 = 2^1 \times 3^2 \times 5^1 $$

Identify the highest powers of each prime factor

Next, we compare the prime factorization results of both numbers.

For the prime factor 2: The highest power is $2^2$
For the prime factor 3: The highest power is $3^2$
For the prime factor 5: The highest power is $5^1$

Calculate the LCM using the highest powers

Now, we multiply the highest powers of all the prime factors together to find the LCM.

$$ LCM = 2^2 \times 3^2 \times 5^1 $$

Calculating this step-by-step:

$$ LCM = 4 \times 9 \times 5 $$ $$ LCM = 36 \times 5 $$ $$ LCM = 180 $$

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

More Information

The LCM is particularly useful in solving problems involving fractions or finding equivalent fractions. Additionally, it can be used in scheduling problems, such as determining when two events will coincide.

Tips

Confusing LCM with the greatest common divisor (GCD). It's crucial to understand that LCM finds the least common multiple, while GCD finds the greatest common divisor of numbers.
Not correctly identifying the highest powers of prime factors. Always ensure you are taking the maximum exponent for each prime when calculating the LCM.

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