The Least Common Multiple (LCM) of 25 and 100 is 100.
The LCM is the smallest positive number that is a multiple of two or more numbers. Here's a breakdown of why the LCM of 25 and 100 is 100:
Understanding the Concept of LCM
The least common multiple of any two or more natural numbers is the number that is the lowest of their common multiples. To find the LCM, we need to identify multiples of both numbers and then determine the smallest one they share.
Methods for Finding the LCM
Listing Multiples
One way to find the LCM is by listing multiples of each number until a common multiple is found:
- Multiples of 25: 25, 50, 75, 100, 125, ...
- Multiples of 100: 100, 200, 300, ...
As you can see, the smallest common multiple is 100.
Prime Factorization Method
Another method involves finding the prime factorization of each number:
- Prime factorization of 25: 5 x 5 (or 52)
- Prime factorization of 100: 2 x 2 x 5 x 5 (or 22 x 52)
To find the LCM, we take the highest power of each prime factor that appears in the factorizations:
- 22 (from 100)
- 52 (from 25 and 100)
Multiplying these together: 22 x 52 = 4 x 25 = 100.
Conclusion
In this particular case, 100 is a multiple of 25, as 25 x 4 = 100. Since 100 is also obviously a multiple of itself, it’s the smallest common multiple of both numbers. The LCM of 25 and 100 is therefore 100.
