1. **Stating the problem:** We need to find the number of ordered positive integer pairs $(a,b)$ such that $$a^2 - b^2 = 2025.$$ 2. **Formula and important rules:** Recall the difference of squares factorization: $$a^2 - b^2 = (a-b)(a+b).$$ Since $a$ and $b$ are positive integers, both $a-b$ and $a+b$ are positive integers, and $a+b > a-b$. 3. **Rewrite the equation:** Using the factorization, we have $$ (a-b)(a+b) = 2025. $$ Let $$x = a-b, \quad y = a+b,$$ so $$xy = 2025$$ with $x,y$ positive integers and $y > x$. 4. **Express $a$ and $b$ in terms of $x$ and $y$:** $$a = \frac{x+y}{2}, \quad b = \frac{y-x}{2}.$$ Since $a,b$ are positive integers, both $\frac{x+y}{2}$ and $\frac{y-x}{2}$ must be positive integers. 5. **Conditions on $x$ and $y$:** - $x$ and $y$ are positive divisors of 2025 with $y > x$. - $x$ and $y$ must have the same parity (both odd or both even) so that $a$ and $b$ are integers. - Since 2025 is odd, all its divisors are odd, so $x$ and $y$ are odd, satisfying the parity condition. 6. **Factorize 2025:** $$2025 = 3^4 \times 5^2.$$ 7. **Count the number of positive divisors:** Number of divisors = $(4+1)(2+1) = 5 \times 3 = 15.$ 8. **List divisors:** The 15 divisors are: $$1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025.$$ 9. **Count factor pairs $(x,y)$ with $xy=2025$ and $y>x$:** There are 15 divisors, so 15 pairs $(x,y)$ with $xy=2025$ if we consider $x \leq y$. Since 2025 is a perfect square? Check if 2025 is a perfect square: $$\sqrt{2025} = 45,$$ so yes. Number of factor pairs is $\frac{15+1}{2} = 8$ pairs: $$(1,2025), (3,675), (5,405), (9,225), (15,135), (25,81), (27,75), (45,45).$$ 10. **Exclude pairs where $x=y$ because $a,b$ must be positive:** For $(45,45)$, $$a = \frac{45+45}{2} = 45, \quad b = \frac{45-45}{2} = 0,$$ which is not positive. So exclude this pair. 11. **Check positivity of $a$ and $b$ for other pairs:** Since $y > x > 0$, $$a = \frac{x+y}{2} > 0, \quad b = \frac{y-x}{2} > 0.$$ All other pairs satisfy this. 12. **Final answer:** There are 7 ordered positive integer pairs $(a,b)$ satisfying $a^2 - b^2 = 2025$.