What is the least common multiple of 25 and 45?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 25 and 45, which involves identifying the smallest number that is a multiple of both given numbers.
Answer
The least common multiple of 25 and 45 is $225$.
Answer for screen readers
The least common multiple (LCM) of 25 and 45 is 225.
Steps to Solve
- Find the prime factorization of each number
To find the LCM, start by determining the prime factorization of both numbers.
- The prime factorization of $25$ is $5^2$ (since $25 = 5 \times 5$).
- The prime factorization of $45$ is $3^2 \times 5^1$ (since $45 = 3 \times 3 \times 5$).
- Identify the highest power of each prime factor
Next, look at the prime factors and their highest powers from both factorizations.
- The prime factor $3$ appears in $45$ with the highest power of $3^2$.
- The prime factor $5$ appears in both, and the highest power is $5^2$.
- Multiply the highest powers together to get the LCM
Now, multiply the highest powers of each prime factor to get the LCM.
- The LCM is calculated as: $$ LCM = 3^2 \times 5^2 $$
- Calculate the final value
Finally, calculate the values:
- $3^2 = 9$
- $5^2 = 25$
- Therefore, $LCM = 9 \times 25 = 225$.
The least common multiple (LCM) of 25 and 45 is 225.
More Information
The LCM is useful in many mathematical applications, including adding fractions with different denominators and solving problems involving cycles or repetitions in real life. The concept of LCM is foundational in number theory and helps us understand the relationship between numbers.
Tips
One common mistake is miscalculating the powers of prime factors or forgetting to consider all distinct primes when finding the LCM. To avoid this, always verify the prime factorizations and the highest powers of each prime.