least common multiple of 10 and 15

2025

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31-03-2025

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The least common multiple (LCM) is the smallest positive integer that is divisible by both numbers. Let's find the LCM of 10 and 15. We'll explore a few methods to solve this.

Method 1: Listing Multiples

The simplest method is to list the multiples of each number until we find the smallest multiple they share.

Multiples of 10: 10, 20, 30, 40, 50, 60...

Multiples of 15: 15, 30, 45, 60, 75...

Notice that 30 is a multiple of both 10 and 15. However, 30 is not the least common multiple. Looking further, we see that both lists contain 30 and 60. The smallest shared multiple is 30. Therefore, the LCM(10, 15) = 30.

This method works well for smaller numbers but can become cumbersome with larger numbers.

Method 2: Prime Factorization

This method uses the prime factorization of each number. Prime factorization is expressing a number as a product of its prime factors.

• Prime factorization of 10: 2 x 5
• Prime factorization of 15: 3 x 5

To find the LCM, we take the highest power of each prime factor present in either factorization and multiply them together.

• We have a '2' in the factorization of 10, and a '3' in the factorization of 15.
• Both numbers share a '5'. We only need to include it once.

Therefore, the LCM(10, 15) = 2 x 3 x 5 = 30

This method is more efficient for larger numbers.

Method 3: Using the Greatest Common Divisor (GCD)

The LCM and the greatest common divisor (GCD) are related. The product of the LCM and GCD of two numbers is equal to the product of the two numbers.

First, let's find the GCD of 10 and 15. The GCD is the largest number that divides both 10 and 15 without leaving a remainder. The factors of 10 are 1, 2, 5, and 10. The factors of 15 are 1, 3, 5, and 15. The greatest common factor is 5.

Now, using the formula:

LCM(a, b) = (a x b) / GCD(a, b)

LCM(10, 15) = (10 x 15) / 5 = 150 / 5 = 30

This method is also efficient, especially when dealing with larger numbers where finding the prime factorization might be more challenging.

Conclusion

We've demonstrated three different methods to find the least common multiple of 10 and 15. All methods arrive at the same answer: 30. Choosing the best method depends on the numbers involved and your comfort level with different mathematical approaches. Understanding the LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cycles or rhythms.