LCM of 75 and 30

Steps to Solve

1. Find Prime Factorization of 75

   The first step is to find the prime factors of 75.

   We can do this by dividing by the smallest prime numbers:

   $$ 75 \div 3 = 25 $$

   Now, factor 25:

   $$ 25 \div 5 = 5 $$

   Finally, we have:

   $$ 5 \div 5 = 1 $$

   Therefore, the prime factorization of 75 is:

   $$ 75 = 3^1 \times 5^2 $$

2. Find Prime Factorization of 30

   Next, we find the prime factors of 30 using a similar method:

   $$ 30 \div 2 = 15 $$

   Now, factor 15:

   $$ 15 \div 3 = 5 $$

   And we have:

   $$ 5 \div 5 = 1 $$

   Thus, the prime factorization of 30 is:

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

3. Identify the Highest Powers of All Prime Factors

   Now we list each prime factor found in both numbers and take the highest power:

   • For the prime factor 2: $2^1$ (from 30)
   • For the prime factor 3: $3^1$ (common in both)
   • For the prime factor 5: $5^2$ (from 75)

4. Calculate the LCM using the prime factors

   To find the LCM, multiply all the highest powers of the prime factors together:

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

   Now compute it:

   $$ \text{LCM} = 2 \times 3 \times 25 $$

   $$ \text{LCM} = 6 \times 25 = 150 $$