LCM of 50 and 30

Steps to Solve

1. Identify the prime factors of each number

For 50, we can divide by the smallest prime number 2:

$$ 50 \div 2 = 25 $$

Next, 25 can be divided by 5:

$$ 25 \div 5 = 5 $$

And finally,

$$ 5 \div 5 = 1 $$

So, the prime factorization of 50 is

$$ 50 = 2^1 \times 5^2 $$

Similarly, for 30, we can divide by 2:

$$ 30 \div 2 = 15 $$

Next, divide 15 by 3:

$$ 15 \div 3 = 5 $$

And finally,

$$ 5 \div 5 = 1 $$

Thus, the prime factorization of 30 is

$$ 30 = 2^1 \times 3^1 \times 5^1 $$

2. Determine the highest power of each prime factor

Next, we take the highest power of each prime factor present in both factorizations:

• For the prime factor 2: the highest power is $2^1$ (from both 50 and 30)
• For the prime factor 3: the highest power is $3^1$ (from 30)
• For the prime factor 5: the highest power is $5^2$ (from 50)

3. Calculate the LCM using the prime factors

We now multiply these highest powers together to find the LCM:

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

Calculating this step-by-step:

First, calculate $5^2 = 25$.

Then multiply:

$$ 2 \times 3 = 6 $$

Now multiply:

$$ 6 \times 25 = 150 $$

Thus, the least common multiple is 150.