We are tasked with finding the smallest value of $m$ such that any collection of $m$ distinct positive integers contains at least two numbers whose sum or difference is divisible by 10. Let's break the problem into manageable steps:

Step 1: Modular Arithmetic Setup

Consider the numbers modulo 10. Every integer $x$ can be written as $x \equiv r \pmod{10}$, where $r \in \{0, 1, 2, \ldots, 9\}$. So, for any collection of integers, their behavior modulo 10 is central to the problem. We are interested in when the sum or difference of two numbers is divisible by 10. This occurs when: $x + y \equiv 0 \pmod{10} \quad \text{or} \quad x - y \equiv 0 \pmod{10}.$

This translates to:

• $x + y \equiv 0 \pmod{10}$ if the remainders of $x$ and $y$ modulo 10 add up to 10 (or 0 for identical remainders).
• $x - y \equiv 0 \pmod{10}$ if $x \equiv y \pmod{10}$, i.e., the remainders are the same.

Step 2: Analyze the Remainders

The numbers modulo 10 are $\{0, 1, 2, \ldots, 9\}$. To ensure that no two numbers in a collection sum or differ by a multiple of 10, we need to avoid:

• Any two numbers having the same remainder modulo 10 (this would give $x - y \equiv 0 \pmod{10}$).
• Any two numbers whose remainders add up to 10, i.e., pairs like $(1, 9), (2, 8), (3, 7), (4, 6), (5, 5)$.

Thus, we need to carefully select remainders such that no two remainders are the same and no two remainders add up to 10.

Step 3: Maximize the Collection Without Violating the Conditions

We will now try to form the largest collection of distinct remainders modulo 10 that satisfies these conditions:

• Choose remainders $0$ and $5$, but we can only select one of each, as adding two $0$'s or two $5$'s would violate the difference condition.
• For the other remainders, avoid pairing remainders that sum to 10. This allows us to choose one element from each pair of remainders:
  • Choose either $1$ or $9$, but not both.
  • Choose either $2$ or $8$, but not both.
  • Choose either $3$ or $7$, but not both.
  • Choose either $4$ or $6$, but not both.

Thus, the largest set of remainders we can select without violating the sum or difference condition consists of 5 elements: $\{0, 1, 2, 3, 4\}.$

Step 4: Conclusion

If we try to include more than 5 numbers, there will be at least two numbers whose sum or difference is divisible by 10. Hence, $m = 6$ is the smallest value for which the condition is violated, since for any collection of 6 distinct numbers modulo 10, at least two of them must either have the same remainder or sum to 10.

Therefore, the smallest value of $m$ is $\boxed{6}$, and this is the best possible value because any collection of 5 numbers can be chosen without violating the conditions, but any collection of 6 will always violate the condition.

Relative Questions:

1. What is the principle of modular arithmetic and how is it applied in this problem?
2. How can the concept of remainder classes modulo 10 be used to solve other similar problems?
3. Can this approach be generalized for sums and differences divisible by numbers other than 10?
4. What happens if we change the number base (e.g., modulo 12 instead of modulo 10)?
5. What alternative strategies can be used to find patterns when dealing with sums and differences in integer sets?

Tip:

Modular arithmetic is a powerful tool when dealing with problems involving divisibility or cyclic patterns in numbers. Always consider simplifying complex conditions with modulo operations to spot hidden symmetries.