91Ó°ÊÓ

In discrete mathematics, proving that an expression is even can be a methodical process. When asked to prove that the expression \(x^{3} y - x y^{3}\) is even for all positive integers \(x\) and \(y\), one effective approach is to factor the expression. By factoring, we get \(x y (x^2 - y^2)\). Here, the key term is the difference of squares, \(x^2 - y^2\). Any difference of squares is inherently even when the squares themselves involve different parity, typically seen in sequences of consecutive integers.

Hence, \(x^2 - y^2\) is even and multiplying it by \(xy\), a product of two positive integers, still results in an even number. This shows that \(x^{3} y - x y^{3} = x y (x^2 - y^2)\) is divisible by 2 and is therefore even. The evenness proof capitalizes on these characteristics of integer algebra, and understanding this factoring can significantly simplify many other mathematical problems involving integer properties.

The construction of graphs is a fascinating aspect of discrete mathematics that involves creating connections, designated as edges, between a set of points referred to as vertices. In this problem, we start with a vertex set \(V = \{1, 2, 3, \ldots, 9\}\). The graph, \(G = (V, E)\), is undirected and loop-free. This means each edge connects two different vertices and there are no multiple edges between the same pair.

To draw an edge between two vertices \(m\) and \(n\), the condition is that either \(5\) divides the sum \(m+n\) or the difference \(m-n\). For instance, if \(m+n\) equals 10, 15, or any multiple of 5, then \(\{m, n\}\) is an edge in the graph. Similarly, if \(m-n\) is divisible by 5, the edge exists. This selective process ensures that the resulting graph reflects specific properties dictated by divisibility, a common theme in combinatorial mathematics.

The problem of proving that given any three distinct positive integers, there are two integers, say \(x\) and \(y\), such that 10 divides \(x^{3} y - x y^{3}\), incorporates several foundational concepts in number theory and modular arithmetic. To solve this, remember the expression \(x^{3} y - x y^{3}\) can be rewritten as \(xy(x^2 - y^2)\), and for divisibility by 10, both 2 and 5 must divide this product.

By the Pigeonhole Principle, since there are three integers and only three possible residue classes modulo 3, at least two integers must be congruent modulo 3. This congruence means their difference, hence \(x^2 - y^2\), is divisible by 3, making it even when dealing with consecutive or appropriately paired integers. Additionally, testing their residues modulo 5, with integers separated into four classes (1, 2, 3, 4), assures that at least two integers \((x, y)\) share the same residue. This fulfills the requirement for divisibility by 5 in the same expression.

Thus, the product \(x^{3}y - x y^{3}\) is divisible by both 2 and 5, leading to overall divisibility by 10. Such problems enhance understanding of the elegant balance between structure (modular classes) and algebraic expression in integers.