Solution: Let $x$ be the integer. Then $x \equiv -1 \pmod{5}$ and $x \equiv -1 \pmod{7}$. This implies $x + 1$ is divisible by both 5 and 7, so $x + 1 = \text{LCM}(5, 7) = 35$. Thus, $x = 34$. $\boxed{34}$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 $?

["Optimizing Robotic Arm Gearing: Solving for $ a^3 + b^3 $", "When designing precision mechanical systems—especially robotic arms—efficient gear ratios and synchronized motion are critical. Consider a real-world scenario where two gears rotate with integer speeds $ a $ and $ b $. Suppose their combined rotational frequency satisfies:\n$$\na + b = 10 \quad \ ext{and} \quad a^2 + b^2 = 58\n$$\nThese constraints reveal a deeper algebraic structure that can be leveraged to compute key performance metrics—such as rotational energy memory or cubic response dynamics. But beyond mechanics, this problem beautifully illustrates how number theory underpins engineering design. Here, we derive $ a^3 + b^3 $ using both identity identities and logical reasoning.", "Let $ x = a $ and $ y = b $. From the given:\n- $ a + b = 10 $ → $ s = 10 $\n- $ a^2 + b^2 = 58 $", "We know the identity:\n$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$\nThus, we need both $ a + b $ and $ ab $ to compute the result. We already have $ a + b = 10 $. To find $ ab $, use the identity:\n$$\n(a + b)^2 = a^2 + 2ab + b^2\n$$\nSubstitute known values:\n$$\n10^2 = 58 + 2ab \Rightarrow 100 = 58 + 2ab \Rightarrow 2ab = 42 \Rightarrow ab = 21\n$$", "Now substitute into the cubic identity:\n$$\na^3 + b^3 = 10^3 - 3 \cdot 21 \cdot 10 = 1000 - 630 = 370\n$$", "Alternatively, since $ a + b = 10 $ and $ ab = 21 $, $ a $ and $ b $ are roots of the quadratic:\n$$\nt^2 - 10t + 21 = 0\n$$\nFactoring:\n$$\n(t - 7)(t - 3) = 0 \Rightarrow t = 7 \ ext{ or } 3\n$$\nSo $ (a, b) = (7, 3) $ or $ (3, 7) $. Compute $ a^3 + b^3 $:\n$$\n7^3 + 3^3 = 343 + 27 = 370\n$$", "This result isn’t just a number—it represents a measurable physical quantity in kinetic energy relationships or motion harmonics across coupled gears. In the context of robotic arm dynamics, such cubic sums may model rotational effort over time or cumulative torque modeled as polynomial functions of speed.", "Final Answer: $ \boxed{370} $", "This elegant solution merges number theory with mechanical engineering insight, demonstrating how mathematical precision enhances robotic design. By solving constraints symbolically and verifying numerically, engineers can confirm optimal configurations efficiently—turning abstract equations into practical innovation."]