1. Let's restate the problem: Find a factor of 300 such that when the factor and some other number are added, the sum equals -65. 2. First, list the factors of 300. Since 300 = $2^2 \times 3 \times 5^2$, the positive factors are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300. 3. Since the sum is negative (-65), one or both numbers must be negative. 4. Suppose the factor of 300 is $f$, then we want $f + x = -65$. 5. To find such pairs where $f$ divides 300, check negative factors as well (i.e., $-1, -2, ..., -300$). 6. Rewrite the equation as $x = -65 - f$. 7. Since $f$ is a factor of 300, check which $f$ (positive or negative) makes $x$ also an integer (no other conditions are specified). 8. Examples: If $f = -60$ (which divides 300), then $x = -65 - (-60) = -5$. Both integers, sum is -65. 9. So one factor is $-60$. Indeed, $-60$ divides 300 because $300 / -60 = -5$ (an integer). 10. Another example: $f = -75$, then $x = -65 - (-75) = 10$. 11. $-75$ divides 300 since $300 / -75 = -4$. 12. Therefore, factors of 300 such as $-60$ or $-75$ satisfy the condition with the added number making the sum $-65$. Final answer: One factor of 300 that works is $\boxed{-60}$ (with the other number being $-5$), or $\boxed{-75}$ (with the other number 10).