Understanding the Least Common Multiple (LCM) of 15, 20, and 30

When searching for the smallest number that is exactly divisible by 15, 20, and 30, one might be tempted to think of zero. However, zero is not the correct answer. It can be divided by any number without leaving a remainder, but it is not the smallest positive integer with this property. The smallest positive integer that precisely meets our criteria is 60. This article will walk you through the process of finding this number and explain the mathematical concept behind it.

What is the Least Common Multiple (LCM)?

The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of the numbers without leaving a remainder. In simpler terms, it's the smallest number that can be evenly divided by a set of given numbers.

Prime Factorization Method

To find the LCM of 15, 20, and 30, we begin by finding the prime factorization of each number:

15 3 × 5 20 22 × 5 30 2 × 3 × 5

Next, we identify the highest power of each prime factor in these factorizations. The prime factors for these numbers are 2, 3, and 5. We take the highest power of each prime factor:

Highest power of 2 is 22 from 20 Highest power of 3 is 31 from 15 and 30 Highest power of 5 is 51 from all three numbers

By multiplying these highest powers together, we can calculate the LCM:

LCM 22 × 3 × 5 4 × 3 × 5 60

Verification through Multiplication

To further confirm that 60 is indeed the LCM, we can check its divisibility by 15, 20, and 30:

60 ÷ 15 4 (no remainder) 60 ÷ 20 3 (no remainder) 60 ÷ 30 2 (no remainder)

This confirms that 60 is the smallest positive integer that can be divided by 15, 20, and 30 without leaving a remainder.

Additional Examples and Considerations

Let's consider a few more examples to solidify our understanding of LCM:

Example 1:

15 3 × 5 20 22 × 5 30 2 × 3 × 5

LCM 22 × 3 × 5 4 × 3 × 5 60

Example 2:

15 3 × 5 21 3 × 7 27 33

The common factor of all these numbers is 3. The LCM is 3 × 5 × 7 × 32 1890. Thus, 1890 is the smallest number divisible by 15, 21, and 27.

Example 3:

15 3 × 5 20 22 × 5 30 2 × 3 × 5

LCM 22 × 3 × 5 60

Example 4:

15 3 × 5 25 52 40 23 × 5

LCM 23 × 3 × 52 600

Conclusion

The least common multiple (LCM) of 15, 20, and 30 is 60. This number is exactly divisible by each of the given numbers without leaving a remainder. Understanding how to calculate the LCM is crucial in various mathematical and practical applications, such as solving algebraic equations, managing time schedules, and simplifying fractions.