How to Find the Smallest Divisible Number from 1 to n

Have you ever stumbled upon a problem where you need to find a number that is divisible by all integers from 1 to a given number (n)? This task might seem daunting at first, but with the right approach, you can easily solve it. In this guide, we will explore how to find the smallest positive whole number divisible by all integers from 1 to (n).

The Method: Finding Prime Factors and Prime Powers

To find the smallest number divisible by all integers from 1 to (n), you need to consider the prime factors of each number in that range. Here’s a step-by-step process to tackle the problem:

Identify Prime Factors and Prime Powers: List out all the prime numbers and prime powers (numbers that are powers of primes) that are less than or equal to (n). These numbers will have the highest power of each prime factor that appears in the range from 1 to (n).Select the Highest Powers: Among the prime factors (including powers of primes), select the highest power that appears for each prime. For example, if 7 is a prime and 4 (which is (2^2)) is a prime power, choose the highest power in the range (which would be 4 in this case).Calculate the Product: Multiply these highest powers together to get the smallest number divisible by all integers from 1 to (n).

Example: Finding the Smallest Divisible Number from 1 to 7

Let’s walk through an example where we need to find the smallest number that is divisible by all integers from 1 to 7.

List Prime Factors and Prime Powers: - Prime numbers: 2, 3, 5, 7 - Prime powers: 4 (which is (2^2))Select the Highest Powers: - For 2: The highest power is 4 (which is (2^2)) - For 3: The highest power is 3 - For 5: The highest power is 5 - For 7: The highest power is 7Calculate the Product: - Multiply these highest powers together: (4 times 3 times 5 times 7 420)

Therefore, 420 is the smallest positive whole number divisible by all integers from 1 to 7.

Understanding Prime Factorization and Least Common Multiple (LCM)

The method described above is closely related to the concept of the least common multiple (LCM). The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set. In this context, the LCM of the numbers from 1 to (n) is the smallest number that is divisible by all of them.

To further illustrate, let’s consider a more complex example. Suppose (n 10): - Prime numbers: 2, 3, 5, 7 - Prime powers: 4 (which is (2^2)) From this, we select the highest powers: - For 2: The highest power is 8 (which is (2^3)) - For 3: The highest power is 3 - For 5: The highest power is 5 - For 7: The highest power is 7 The product of these highest powers is: (8 times 3 times 5 times 7 840)

Thus, 840 is the smallest number divisible by all integers from 1 to 10.

Conclusion

Finding the smallest divisible number from 1 to (n) involves a systematic approach that considers the prime factors and prime powers within the range. By selecting the highest powers of each prime and multiplying them, you can easily determine the LCM of the numbers from 1 to (n). This method simplifies a complex problem into a straightforward calculation, making it a valuable tool in many mathematical and real-world applications.

References

For a deeper understanding, refer to the following resources: - [Mathematics Stack Exchange]() - [Khan Academy: Least Common Multiple]()) - [Wikipedia: Least Common Multiple](_common_multiple)