lcm of 45 and 75

Steps to Solve

1. Finding the Prime Factorization of Each Number

First, we need to break down the numbers 45 and 75 into their prime factors.

• The prime factorization of 45:
  • 45 can be divided by 3: $45 \div 3 = 15$
  • 15 can also be divided by 3: $15 \div 3 = 5$
  • Finally, 5 is a prime number.

Thus, the prime factorization of 45 is: $$ 45 = 3^2 \times 5^1 $$

• The prime factorization of 75:
  • 75 can be divided by 3: $75 \div 3 = 25$
  • 25 can be divided by 5: $25 \div 5 = 5$
  • 5 is a prime number.

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

2. Identifying the Highest Powers of Each Prime Factor

Next, we will identify the highest powers of each prime factor in both factorizations.

• From 45:

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

• From 75:

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

The highest powers we have are:

• For 3: $3^2$
• For 5: $5^2$

3. Calculating the LCM

Now we can calculate the LCM by multiplying the highest powers of all prime factors together.

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

Calculating this gives: $$ 3^2 = 9 $$ $$ 5^2 = 25 $$ Then, $$ LCM = 9 \times 25 = 225 $$