Probability of Randomly Selecting a Number Divisible by 810 or 12 within 1-1000

Many interesting mathematical problems revolve around the concept of divisibility and probabilities. For example, let's explore the question of finding the probability of picking a number between 1 and 1000 that is divisible by 810 or 12. This problem involves an understanding of Least Common Multiple (LCM) and fundamental principles of probability.

The Role of LCM in Divisibility

To solve problems related to divisibility by more than one number, it is useful to understand the concept of the Least Common Multiple (LCM). The LCM of two numbers is the smallest positive integer that is divisible by both the numbers. For instance, the LCM of 8 and 10 can be found by considering their prime factorizations:

8 23

10 2 x 5

The LCM of 8 and 10 is therefore 23 x 5 40. This means any number that is a multiple of 40 will be divisible by both 8 and 10.

Probability of Selecting a Number Divisible by 8 and 10

Given a range of 1 to 1000, we need to determine how many numbers in this range are multiples of 40. The calculation is straightforward:

Number of multiples of 40 in the range [1, 1000] 1000 / 40 25

The probability of randomly selecting a number divisible by both 8 and 10 is therefore:

Probability Number of favorable outcomes / Number of possible outcomes

Probability 25 / 1000 0.025 or 2.5%

Extending the Concept to 810 or 12

Now, let's extend our exploration to find the probability of picking a number between 1 and 1000 that is divisible by either 810 or 12.

Firstly, we need to check the prime factors of 810 and 12:

810 2 × 34 × 5

12 22 × 3

Since we are looking for numbers that might be divisible by 810 or 12, we need to consider their LCM. The LCM of 810 and 12 includes all prime factors from both numbers with the highest powers:

LCM(810, 12) 22 × 34 × 5 3240

Given that 3240 is greater than 1000, it is clear that no number between 1 and 1000 can be divisible by 3240. Therefore, the only relevant divisibility in this range is for 12.

The prime factors of 12 are:

12 22 × 3

This means that any number divisible by 12 between 1 and 1000 must be a multiple of 12. The count of multiples of 12 in this range is:

1000 / 12 83.33 (approximately 83)

Hence, the probability of randomly selecting a number divisible by 12 within the given range is:

Probability 83 / 1000 0.083 or 8.3%

Conclusion

In conclusion, the probability of picking a number between 1 and 1000 that is divisible by both 8 and 10 is 2.5%, while the probability for a number divisible by 810 or 12 is 8.3%.

Key Takeaways

Understanding the Least Common Multiple (LCM) is crucial for determining divisibility. The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes. In the given range, numbers divisible by 810 or 12 have higher probabilities due to the LCM calculations.

This analysis highlights the importance of LCM in solving divisibility and probability problems and demonstrates how prime factorization plays a key role in determining the LCM between two numbers.