The Smallest Number Divisible by 15, 20, and 24: Finding the Least Common Multiple (LCM)

Understanding the concept of the smallest number that can be divided exactly by 15, 20, and 24 involves delving into the concept of the least common multiple (LCM). Let's explore this topic in detail.

Introduction to Least Common Multiple (LCM)

The LCM of a set of numbers is the smallest number that can be divided by each of the numbers in the set without leaving a remainder. For the numbers 15, 20, and 24, finding their LCM will help us identify the smallest number that can be exactly divided by all three.

Method for Finding the LCM

To find the LCM of 15, 20, and 24, follow these steps:

Prime Factorization

15: (15 3 times 5) 20: (20 2^2 times 5) 24: (24 2^3 times 3)

Determine the Highest Powers of Each Prime

For (2): The highest power is (2^3) from 24. For (3): The highest power is (3^1) from 15 and 24. For (5): The highest power is (5^1) from 15 and 20.

Calculate the LCM

The LCM is found by multiplying the highest powers of all prime factors.

(text{LCM} 2^3 times 3^1 times 5^1)

Calculating this step-by-step:

(2^3 8) (3^1 3) (5^1 5)

Now multiply these together:

(8 times 3 24) (24 times 5 120)

Thus, the smallest number that can be divided exactly by 15, 20, and 24 is 120.

Further Examples and Problems

Let's explore some additional examples to solidify our understanding:

Example 1: Split 10000 into prime factors: 15: (15 3 times 5) 20: (20 2^2 times 5) 25: (25 5^2)

The LCM is 300: (2^2 times 3 times 5^2).

The smallest 5-digit number is 10000. Dividing 10000 by 300, we get (10000 / 300 33.333ldots).

So, the smallest 5-digit number exactly divisible by 15, 20, and 25 is 34 times; 300 10200.

Conclusion

Understanding the LCM is fundamental in solving problems related to divisibility and factorization. The process involves prime factorization, identifying the highest powers of each prime, and then multiplying these to find the LCM. By practicing with different sets of numbers, one can improve their skills in finding the LCM and applying it to various real-world scenarios.

Frequent Questions and Answers

What is the LCM of 15, 21, and 27? What is the LCM of 15, 21, and 30? What is the LCM of 15, 20, and 40?

These problems can be solved using the same method of prime factorization and identifying the highest powers of each prime factor, multiplying them to obtain the LCM. The answers are as follows:

LCM of 15, 21, and 27: 15: (3 times 5) 21: (3 times 7) 27: (3^3) LCM 3^3 × 5 × 7 315 LCM of 15, 21, and 30: 15: (3 times 5) 21: (3 times 7) 30: (2 times 3 times 5) LCM 2 × 3^2 × 5 × 7 210 LCM of 15, 20, and 40: 15: (3 times 5) 20: (2^2 times 5) 40: (2^3 times 5) LCM 2^3 × 3 × 5 120