Which smallest number must be added to 710 so that the sum is a perfect cube?

Concept & Logic The core logic is to find the nearest perfect cube that is strictly greater than the given number. Step-by-Step Solution * Identify the closest perfect cubes around 710. * We know that $8^3 = 512$ and $9^3 = 729$. * Since $729 > 710$, the next perfect cube immediately after 710 is 729. * To find the smallest number to add, subtract the given number from the perfect cube: $729 - 710 = 19$. Exam Strategy & Shortcut Memorize perfect cubes up to 15. Recognizing $9^3 = 729$ instantly gives the difference without any manual calculation. Common Pitfall A common mistake is finding the nearest perfect square instead of a cube, or subtracting instead of adding. Always read carefully whether the question asks to "add" or "subtract". Final Answer Therefore, the correct answer is **19**.