91Ó°ÊÓ

[This problem requires a mixture of serious thought and written proof.] (a) I choose six integers between 10 and 19 (inclusive). (i) Prove that some pair of integers among my chosen six must be relatively prime. (ii) Is it also true that some pair must have a common factor? (b) I choose six integers in the nineties (from \(90-99\) inclusive). (i) Prove that some pair among my chosen integers must be relatively prime. (ii) Is it also true that some pair must have a common factor? (c) I choose \(n+1\) integers from a run of \(2 n\) consecutive integers. (i) Prove that some pair among the chosen integers must be relatively prime. (ii) Is it also true that some pair must have a common factor?

Step by step solution

01

Understand the Problem

We have three different subproblems involving choosing a set of integers within a specific range and analyzing their properties, focusing on relatively prime pairs and those with common factors.

02

Focus on Problem (a)(i)

Select six integers between 10 and 19. These numbers are between 10 and 19 inclusive, giving us 10 possible numbers: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. To prove that some pair of integers among my chosen six must be relatively prime, notice that directly consecutive pairs, such as (11,12), are often relatively prime. Notably, (11, 12) is indeed relatively prime because 11 is prime, and 12 is not a multiple of 11.

03

Problem (a)(ii) Analysis

We need to prove that some pair has a common factor. Take any even number like 10 and any other even number like 12; both are divisible by 2, hence sharing a common factor.

04

Problem (b)(i) for Numbers in the Nineties

Select six integers from 90 to 99. For any six numbers within this range, consider the set of numbers: 91, 92, 93, 94, 95, 96, 97, 98, 99. Since 97 is a prime number, picking it along with any non-multiple of 97 assures a relatively prime pair.

05

Problem (b)(ii) for Common Factor in the Nineties

Many numbers are even (e.g., 92, 94, 96, 98), and any pair of even numbers will have 2 as a common factor. So, yes, some pair has a common factor.

06

Problem (c)(i) for Consecutive Integers

Consider choosing n+1 integers from a sequence of 2n consecutive integers, say from 1 to 2n. Consecutive numbers, such as (1, 2) or any similar sequence, ensure that at least one pair like (x,x+1) will be relatively prime.

07

Problem (c)(ii) for Common Factor in Consecutive Integers

In 2n consecutive numbers, some could be multiples of a particular number (e.g., two even numbers), so you can find pairs that share a common factor.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relatively Prime Integers

Relatively prime integers are two numbers that share no common factors other than 1. This means they do not have any other positive integer besides 1 that can divide both of them without leaving a remainder.
A useful way to find relatively prime pairs is by checking consecutive integers, where one number is often a prime while the other is a non-multiple of the former.
For example:

• The numbers 11 and 12 are relatively prime because 11 is a prime number, and 12 cannot be divided by 11 without a remainder.
• In general, if you pick enough numbers from any sequence, like six numbers from 10 to 19 or the nineties, you are likely to find a relatively prime pair by concentrating on one prime number and another which is not a multiple.

Understanding relatively prime integers helps in solving number theory problems where determining the absence of common factors is crucial.

Common Factors

Common factors occur when two numbers have a positive integer greater than 1 that divides both numbers completely. Recognizing common factors is important in problems where you need to identify shared divisors.
For instance:

• When numbers are chosen randomly, such as 10 and 12, you find they both share the common factor 2, since both numbers are divisible by 2.
• In ranges like the nineties, pairs like 92 and 94 are even, hence, both are divisible by 2.

This indicates that in any odd-numbered set of integers, focusing on even numbers is a straightforward way to find common factors. So, whenever dealing with sets of integers, especially consecutive ones, look for simplest shared divisors to find common factors easily.

Consecutive Integers

Consecutive integers are numbers that follow each other in order. These are like 10, 11, 12,... and so on. They play a crucial role in number theory because they automatically satisfy certain properties.
Consecutive integers include both odd and even numbers, creating natural situations where some numbers are bound to be relatively prime, especially when one is an odd or a prime number.
For example:

• Choosing any number, say 15, and the next one, 16, you'll find them to be relatively prime as 15 does not share a divisor with 16 other than 1.
• If you choose numbers like 2 and 3, they are again relatively prime, and you find such pairs repeatedly across any run of consecutive integers.

Hence, when dealing with consecutive integers, you'll often find relatively prime pairs by default, yet also can easily identify common factors, especially when even numbers are involved.