Question

Select 100 integers from the integers \(1,2, \ldots, 200\) such that no one of the chosen values is divisible by any other chosen value. Show that if one of the 100 integers chosen from \(1,2, \ldots, 200\) is less than \(16,\) then one of those 100 numbers is divisible by another.

Short answer

Choosing an integer less than 16 creates unavoidable divisibility conflicts.

Step-by-step solution

Understand the Problem

We need to select 100 integers from the range of 1 to 200 such that no chosen integer divides another. Additionally, we need to show that selecting an integer less than 16 violates this condition.

Identify Potential Divisors

Within any group of integers that includes a range from 1 to a larger integer, smaller numbers are often divisors of the larger numbers. Therefore, any integer less than 16 will have a higher potential to divide another because many multiples of numbers less than 16 exist within 1 to 200.

Consider Multiples of Small Numbers

Any integer less than 16 will have several multiples within the range. For instance, 2 has multiples like 4, 8, 16, ..., and 15 has 30, 45, 60, and so on up to 200. These multiples cannot all be selected due to divisibility constraints.

Attempt Maximum Non-Divisibility Selection

Try selecting integers greater than or equal to 16, considering each subsequent number carefully to prevent divisibility conflicts. If we choose solely from numbers like 101 to 200, we avoid most divisibility since these are not multiples of numbers less than 16.

Prove the Divisibility Conflict

If any number less than 16 is chosen, it has numerous multiples up to 200, automatically reducing available choices that are not divisible by smaller selected numbers. This leads to a scenario where 100 non-divisible integers cannot be selected while including numbers less than 16.

Conclusion

Thus, choosing an integer less than 16 leads to inevitable divisibility within the selected 100 integers, confirming the problem statement.

Key concepts

Integer Selection

In discrete mathematics, integer selection involves picking certain numbers from a given set of numbers under specific conditions. In our exercise, we were tasked with selecting 100 integers from the numbers ranging from 1 to 200. However, there was a key condition: no chosen integer should be divisible by another. This constraint limits the possibilities for selection significantly, as it requires careful analysis to ensure that our selected set respects these rules. When attempting to choose such a set of integers, it's crucial to consider relationships between numbers, especially focusing on divisibility, which directly impacts how many numbers can fit within your criteria. Integer selection problems like these frequently appear in mathematical competitions and tests, showcasing the depth of problem-solving skills and understanding required, making them excellent practice for enhancing logical reasoning.

Divisibility Rules

Divisibility rules help determine if one integer is divisible by another. These rules are fundamental in tackling problems where divisibility is a primary condition. In the context of our problem, selecting integers while avoiding divisibility among them, it's essential to recognize that an integer is divisible by another if there exists a whole number that can multiply with the smaller integer to result in the larger one. For instance, 4 divides 16 because 4 multiplied by 4 equals 16. Numbers less than 16 were highlighted as problematic due to their capability to have multiple occurrences in any list within 1 to 200. Therefore, avoiding numbers below 16 is an effective strategy to prevent any two numbers in our selected set from dividing each other. By applying divisibility rules, one can carefully construct a set of integers that strictly adheres to the non-divisibility rule required in the exercise.

Set Theory

Set theory serves a pivotal role in understanding how collections of numbers (or objects) interact and are formulated under various conditions. In this exercise, the set of integers from 1 to 200 can be considered as our universal set. From this, we are tasked with creating a subset of 100 integers that do not divide each other. The interplay between subsets and the universal set is where the challenge lies. Each choice affects the subsequent potential choices, requiring a strategic approach. By leveraging set theory, one can visualize the relationships more clearly, using concepts like intersection (common elements of sets), union (combining of different sets), and complements (elements not in a set) to comprehend the limitations and possibilities in forming such subsets. Understanding set theory fundamentals allows students to better tackle complex discrete math problems, enabling them to develop well-reasoned strategies for integer selection while meeting given conditions.

Combinatorial Problems

Combinatorial problems often involve finding an optimal arrangement or selection from a discrete set of items, subject to certain restrictions. In our integer selection challenge, we approached a combinatorial problem with the restriction that none of the integers selected can divide another. Combinatorial considerations must be made, factoring in the number of possible selections. How do we ensure the largest selection under the condition that no two numbers are divisible? One solution was to focus on integers from 16 and upwards. This decision was influenced by eliminating potential divisibility by excluding numbers below 16. Such an approach highlights not just the importance of considering single constraints, but also how intersecting constraints can heavily impact outcomes. Practicing combinatorial problems sharpens one's ability to analyze multiple conditions and deduce possible solutions effectively.