Solution: We are tasked with finding the number of integer solutions $(x, y)$ to the equation $ x^2 - y^2 = 2025 $. This can be factored as:

["Solution: Finding Integer Solutions to $ x^2 - y^2 = 2025 $ – A Factored Approach", "The equation $ x^2 - y^2 = 2025 $ is a classic Diophantine equation based on the difference of squares. It offers an elegant algebraic pathway to determine the number of integer solutions $ (x, y) $ without resorting to exhaustive search. In this article, we explore how factoring enables us to efficiently find all integer solutions to this equation.", "---", "### Understanding the Difference of Squares", "Recall the identity:\n$$\nx^2 - y^2 = (x - y)(x + y)\n$$\nSo the equation becomes:\n$$\n(x - y)(x + y) = 2025\n$$\nLet:\n- $ a = x - y $\n- $ b = x + y $", "Then:\n$$\nab = 2025 \quad \ ext{and} \quad x = \frac{a + b}{2}, \quad y = \frac{b - a}{2}\n$$\nFor $ x $ and $ y $ to be integers, both $ a + b $ and $ b - a $ must be even — meaning $ a $ and $ b $ must have the same parity (both odd or both even).", "---", "### Step 1: Prime Factorization of 2025", "We begin by factoring $ 2025 $:\n$$\n2025 = 25 \ imes 81 = 5^2 \ imes 3^4\n$$\nThus, $ 2025 = 3^4 \ imes 5^2 $", "The total number of positive divisors is:\n$$\n(4+1)(2+1) = 15\n$$\nSo $ 2025 $ has 15 positive divisors and 15 corresponding negative divisors, making 30 total integer divisors.", "---", "### Step 2: Finding Factor Pairs $ (a, b) $ Such That $ ab = 2025 $", "Each factor pair $ (a, b) $ of $ 2025 $ gives a candidate solution. Since $ a $ and $ b $ must be both odd or both even to yield integer $ x, y $, and $ 2025 $ is odd, all divisors are odd. Therefore, every factor pair $ (a, b) $ consists of odd integers, satisfying the parity condition.", "There are 30 total integer factor pairs $ (a, b) $ such that $ ab = 2025 $ — 15 positive and 15 negative.", "For each such pair $ (a, b) $, we compute:\n$$\nx = \frac{a + b}{2}, \quad y = \frac{b - a}{2}\n$$\nSince $ a $ and $ b $ are both odd, $ a + b $ and $ b - a $ are even, so $ x, y $ are integers.", "---", "### Step 3: Counting Distinct Integer Solutions", "Each valid pair $ (a, b) $ with $ ab = 2025 $ produces a unique solution $ (x, y) $. Since there are 30 such ordered factor pairs (including negative pairs), we now determine how many distinct integer solutions this generates.", "However, note that different factor pairs $ (a,b) $ may yield the same $ (x, y) $. But because each divisor $ a $ of $ 2025 $ determines a unique $ b = 2025/a $, and since the mapping $ a \mapsto (x,y) $ is bijective over the valid factor pairs (due to distinct $ x, y $ for distinct $ (a,b) $), the number of integer solutions is exactly the number of such valid factorizations.", "But we must be careful: although there are 30 divisor pairs, some may produce duplicate $ (x,y) $. However, since $ x $ and $ y $ depend strictly on $ a $ and $ b $, and every divisor generates a unique $ (x,y) $, we conclude:", "> The total number of integer solutions $ (x, y) $ is equal to the number of integer divisors of 2025, because each divisor $ a $ determines a unique pair $ (a, 2025/a) $, and hence a unique solution.", "There are 30 integer divisors of $ 2025 $, hence 30 integer solutions $ (x, y) $.", "---", "### Step 4: Verifying with Example", "Take $ a = 1 $, $ b = 2025 $:\n$ x = \frac{1 + 2025}{2} = 1013 $, $ y = \frac{2025 - 1}{2} = 1012 $: valid integer solution.", "Take $ a = -1 $, $ b = -2025 $:\n$ x = \frac{-1 - 2025}{2} = -1013 $, $ y = \frac{-2025 + 1}{2} = -1012 $: valid.", "All such pairs produce distinct $ (x,y) $, and symmetry ensures no overlaps.", "---", "### Conclusion", "By leveraging the difference of squares identity and analyzing factor pairs $ (a, b) $ of 2025, we find that every divisor pair leads to a valid integer solution. Since $ 2025 $ has 30 integer divisors, there are exactly:", "$$\n\boxed{30}\n$$", "integer solutions $ (x, y) $ to the equation $ x^2 - y^2 = 2025 $.", "This elegant algebraic solution method shows how factoring transforms a nonlinear Diophantine equation into a manageable search over divisors — a powerful strategy in number theory and algebraic problem solving.", "---", "Keywords:\nDiophantine equation, integer solutions, $ x^2 - y^2 = 2025 $, difference of squares, factor pairs, divisor counting, algebraic factorization, number theory.", "Meta Description:\nExplore how factoring $ x^2 - y^2 = 2025 $ enables efficient computation of all integer solutions via divisor pairs — a powerful technique in solving Diophantine equations efficiently."]