Solution: Let $x$ be the integer. Then $x \equiv -1 \pmod5$ and $x \equiv -1 \pmod7$. This implies $x + 1$ is divisible by both 5 and 7, so $x + 1 = \textLCM(5, 7) = 35$. Thus, $x = 34$. $\boxed34$Question: A retired engineer designing a robotic arm observes that the sum of two gear rotation speeds $ a $ and $ b $ is 10, and the sum of their squares is 58. What is $ a^3 + b^3 $? - Appfinity Technologies