The difference between two integers is 5. Their product is 500. Find the numbers.

## Concept & Logic This problem can be represented algebraically as $x - y = 5$ and $xy = 500$. This leads to a quadratic equation. However, since the problem asks to find both numbers and they are provided as pairs in the options, this is a prime candidate for back-solving. ## Step-by-Step Solution * **Given:** Two integers $x$ and $y$. Difference $x - y = 5$. Product $xy = 500$. * **Calculation / Deduction:** * Let the smaller integer be $y$. The larger integer is $y + 5$. * Their product is $y(y + 5) = 500$. * Expand to a quadratic equation: $y^2 + 5y - 500 = 0$. * We need two numbers that multiply to $-500$ and add to $5$. Those numbers are $25$ and $-20$. * Factor the equation: $(y + 25)(y - 20) = 0$. * So, $y = -25$ or $y = 20$. * Assuming positive integers based on the options, if $y = 20$, then the other number is $20 + 5 = 25$. * The numbers are $20$ and $25$. ## Exam Strategy & Shortcut **Option Elimination:** Do not write any equations. Just test the options to see which pair has a difference of 5 and a product of 500. * (a) 15, 20: Difference is 5. Product is $15 \times 20 = 300$ (Incorrect). * (b) 20, 25: Difference is 5. Product is $20 \times 25 = 500$ (Correct!). This takes literally 3 seconds. ## Common Pitfall Getting tunnel vision and automatically grinding through the quadratic equation $(y^2 + 5y - 500 = 0)$ consumes unnecessary time. In objective exams, if the variables you are solving for are in the options, plug them in. ## Final Answer **Therefore, the correct answer is 20, 25.**