lcm of 75 and 45
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 75 and 45. To find the LCM, we can use the prime factorization method or the listing multiples method.
Answer
$225$
Answer for screen readers
The least common multiple (LCM) of 75 and 45 is $225$.
Steps to Solve
- Find the prime factorization of each number
First, we will find the prime factorization of 75 and 45.
For 75:
- 75 can be divided by 3: $$ 75 \div 3 = 25 $$
- Then, 25 can be divided by 5: $$ 25 \div 5 = 5 $$
- Finally, 5 is a prime number.
So, the prime factorization of 75 is: $$ 75 = 3^1 \times 5^2 $$
For 45:
- 45 can be divided by 3: $$ 45 \div 3 = 15 $$
- Then, 15 can also be divided by 3: $$ 15 \div 3 = 5 $$
- Finally, 5 is prime.
So, the prime factorization of 45 is: $$ 45 = 3^2 \times 5^1 $$
- Identify the highest power of each prime factor
Next, we will identify the highest power of each prime factor from both factorizations:
- For 3: the highest power is $3^2$ from 45.
- For 5: the highest power is $5^2$ from 75.
- Multiply the highest powers together
Now, we will calculate the LCM by multiplying the highest powers of all prime factors: $$ LCM = 3^2 \times 5^2 $$
Calculating this gives: $$ LCM = 9 \times 25 = 225 $$
The least common multiple (LCM) of 75 and 45 is $225$.
More Information
The least common multiple is useful in solving problems that involve adding or subtracting fractions, as well as scheduling events that repeat at different intervals. The LCM helps to find a common timeframe where all events align.
Tips
- Forgetting to include all the prime factors when calculating LCM. Always check that you’ve captured the highest power of each prime from both numbers.
- Confusing LCM with the greatest common divisor (GCD). Make sure to differentiate between the two operations.
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