Let $ a = x - y $ and $ b = x + y $. Then $ ab = 2025 $, and $ x = \frac{a + b}{2} $, $ y = \frac{b - a}{2} $. For $ x $ and $ y $ to be integers, $ a + b $ and $ b - a $ must both be even, which implies $ a $ and $ b $ must be both even or both odd.

["Transforming Variables: How to Ensure Integer Solutions Using Substitutions", "In algebra and problem-solving, substitutions play a powerful role in simplifying equations and revealing deeper structure. One classic identity involves defining:", "[\na = x - y \quad \ ext{and} \quad b = x + y\n]", "This substitution is especially useful when given a product such as ( ab = 2025 ), and the goal of finding integer values for ( x ) and ( y ). Let’s explore how this transformation enables us to solve for ( x ) and ( y ), and what mathematical condition ensures they are integers.", "---", "### From Substitution to Key Insight", "Starting from the definitions:", "[\na = x - y, \quad b = x + y\n]", "We can solve for ( x ) and ( y ) by adding and subtracting these equations:", "[\nx = \frac{a + b}{2}, \quad y = \frac{b - a}{2}\n]", "For ( x ) and ( y ) to be integers, both ( a + b ) and ( b - a ) must be even numbers. This means ( a ) and ( b ) must be both even or both odd. In other words, ( a ) and ( b ) must have the same parity.", "---", "### Analyzing the Product Constraint", "We are given:", "[\nab = 2025\n]", "Now, consider the parity of factors. The number 2025 is odd, being ( 45^2 = (9 \cdot 5)^2 = (3^2 \cdot 5)^2 ). Since it has no factor of 2, every factor pair ( (a, b) ) consists of both odd integers. There are no even factor pairs.", "Therefore, ( a ) and ( b ) are guaranteed to be both odd, which satisfies the condition for ( x ) and ( y ) to be integers.", "---", "### Computing Integer Solutions for ( x ) and ( y )", "We can now find all integer solutions for ( x ) and ( y ) by listing the positive factor pairs of 2025, then applying:", "[\nx = \frac{a + b}{2}, \quad y = \frac{b - a}{2}\n]", "Note that factor pairs come in two forms: ( (a, b) ) and ( (b, a) ), but since ( x ) and ( y ) can switch roles (or signs), consider both.", "Primely, factor 2025:", "[\n2025 = 3^4 \cdot 5^2\n]", "So it has ( (4+1)(2+1) = 15 ) positive divisors, leading to 15 factor pairs ( (a, b) ) such that ( ab = 2025 ):", "[\n(1, 2025), (3, 675), (5, 405), (9, 225), (15, 135), (25, 81), (27, 75), (45, 45), \ ext{and reverses}\n]", "For each, ( a + b ) and ( b - a ) are both even → ( x, y \in \mathbb{Z} )", "For example:", "- ( a = 45, b = 45 ):\n ( x = \frac{45 + 45}{2} = 45, \quad y = \frac{45 - 45}{2} = 0 )\n- ( a = 27, b = 75 ):\n ( x = \frac{27 + 75}{2} = 51, \quad y = \frac{75 - 27}{2} = 24 )", "Each valid pair produces integer ( x, y ).", "---", "### Special Considerations", "- If ( a = b ), we get ( y = 0 ), still an integer.\n- Negative factors are also valid (e.g., ( (-1, -2025) )), producing other integer solutions.\n- The symmetry of the equation ( ab = 2025 ) ensures all factor pairs yield insightful integer outcomes.", "---", "### Conclusion", "The transformation ( a = x - y ), ( b = x + y ) elegantly converts a two-variable relationship into a symmetric sum-product form. The key insight — that ( x ) and ( y ) are integers iff ( a ) and ( b ) are both even or both odd — is total satisfied, since ( ab = 2025 ) is odd, forcing ( a ) and ( b ) to be odd.", "This problem illustrates how substitution and parity reasoning unlock structure in algebraic identities — a valuable skill in mathematics, programming, and algorithm design involving integer solutions.", "---", "Keywords:\nLet ( a = x - y ), ( b = x + y ), integer solutions, parity condition, ( ab = 2025 ), how to ensure ( x, y ) are integers, algebraic substitution, Diophantine equations, algebra tips", "Meta Description:\nExplore how substituting ( a = x - y ) and ( b = x + y ) simplifies solving ( ab = 2025 ), and understand the condition that ensures ( x ) and ( y ) are integers. Learn why ( a ) and ( b ) must both be odd."]