LCM of 45 and 25
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 45 and 25. This involves finding the smallest positive integer that is divisible by both numbers.
Answer
The LCM of 45 and 25 is 225.
Answer for screen readers
The least common multiple (LCM) of 45 and 25 is 225.
Steps to Solve
- Finding the Prime Factorization of 45 First, we need to break 45 down into its prime factors. We can do this by dividing by the smallest prime number:
$$ 45 = 5 \times 9 = 5 \times 3^2 $$
So, the prime factorization of 45 is $5^1 \times 3^2$.
- Finding the Prime Factorization of 25 Next, let's find the prime factorization of 25:
$$ 25 = 5 \times 5 = 5^2 $$
So, the prime factorization of 25 is $5^2$.
- Identifying the Maximum Exponents of Each Prime Factor Now, we will list the prime factors we have from both numbers and take the highest power of each:
- For prime factor 3: The maximum exponent is $2$ from 45.
- For prime factor 5: The maximum exponent is $2$ from 25.
- Calculating the LCM To find the LCM, we multiply the prime factors raised to their highest exponents:
$$ \text{LCM} = 3^2 \times 5^2 $$
Calculating the values:
$$ = 9 \times 25 $$
- Final Calculation Now we compute:
$$ 9 \times 25 = 225 $$
Thus, the least common multiple of 45 and 25 is 225.
The least common multiple (LCM) of 45 and 25 is 225.
More Information
The least common multiple is useful in various applications such as solving problems involving fractions, scheduling, and determining common periods for repeating events.
Tips
- Confusing LCM with the greatest common divisor (GCD); remember, LCM is about the least value while GCD is about the greatest common factor.
- Incorrectly factoring numbers; double-check prime factorizations to ensure accuracy.
