lcm of 30 and 75

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 30 and 75. To find the LCM, we can use the prime factorization method or the listing multiples method, among other techniques.

Answer

The least common multiple of 30 and 75 is $150$.

Steps to Solve

Prime Factorization of 30 First, we will find the prime factorization of 30.

30 can be divided by 2: $$ 30 \div 2 = 15 $$
Then, factor 15, which can be divided by 3: $$ 15 \div 3 = 5 $$
Lastly, 5 is already a prime number.

So, the prime factorization of 30 is: $$ 30 = 2^1 \times 3^1 \times 5^1 $$

Prime Factorization of 75 Next, we will find the prime factorization of 75.

75 can be divided by 3: $$ 75 \div 3 = 25 $$
Then, factor 25, which can be divided by 5: $$ 25 \div 5 = 5 $$
Lastly, 5 is already a prime number.

So, the prime factorization of 75 is: $$ 75 = 3^1 \times 5^2 $$

Identifying the Highest Powers of Each Prime Next, we list out the primes from both factorizations:

From 30: $2^1$, $3^1$, $5^1$
From 75: $3^1$, $5^2$

Now, we take the highest power of each prime:

For $2$, the highest power is $2^1$.
For $3$, the highest power is $3^1$.
For $5$, the highest power is $5^2$.

Calculating the LCM Now we'll multiply these highest powers together to find the LCM: $$ LCM = 2^1 \times 3^1 \times 5^2 $$ Calculating this gives: $$ LCM = 2 \times 3 \times 25 = 150 $$

The least common multiple of 30 and 75 is $150$.

More Information

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. In this case, $150$ is the first number that both $30$ and $75$ can divide without leaving a remainder.

Tips

Forgetting to include all prime factors in the LCM calculation.
Failing to choose the highest power of each prime from the factorizations.

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