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Home » 10 Surprising Habits That Lead to a Happy Retirement You’ll Wish You Started Today! - Groen Casting

10 Surprising Habits That Lead to a Happy Retirement You’ll Wish You Started Today! - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: What is the greatest common divisor of $2^{12} - 1$ and $2^{18} - 1$?

📌 Solution: Use the identity $\gcd(2^a - 1, 2^b - 1) = 2^{\gcd(a, b)} - 1$. Compute $\gcd(12, 18) = 6$. Thus, $\gcd(2^{12} - 1, 2^{18} - 1) = 2^6 - 1 = 64 - 1 = \boxed{63}$.

📌 Question: What is the smallest three-digit number divisible by 11 and 13?

📌 Solution: The LCM of 11 and 13 is $143$. The smallest three-digit multiple of 143 is $143 \times 1 = 143$ (since $143 \times 2 = 286$ is also three-digit, but 143 is smaller). Thus, the answer is $\boxed{143}$.Question: Two marine biologists, Dr. Coral and Dr. Tide, arrive at a research station at random times between 7:00 and 8:00. If Dr. Coral arrives after Dr. Tide, what is the probability that Dr. Tide arrived before 7:45?

📌 Solution: Let the arrival times of Dr. Tide and Dr. Coral be $ x $ and $ y $, respectively, where $ x, y \in [0, 60] $ minutes after 7:00. The condition $ y > x $ defines a triangular region in the $ xy $-plane with area $ \frac{60^2}{2} = 1800 $. The favorable region is where $ x < 45 $ and $ y > x $. Integrating over $ x \in [0, 45] $, the area is $ \int_{0}^{45} (60 - x) \, dx = 60 \cdot 45 - \frac{45^2}{2} = 2700 - 1012.5 = 1687.5 $. The conditional probability is $ \frac{1687.5}{1800} = \frac{3}{4} $.

📌 \boxed{\dfrac{3}{4}}

📌 Question: A historian studies three scientific breakthroughs from a collection of 20, where 5 are pivotal. What is the probability that exactly two of the selected breakthroughs are pivotal?

📌 Solution: The total number of ways to choose 3 breakthroughs is $ \binom{20}{3} = 1140 $. The number of favorable outcomes is $ \binom{5}{2} \cdot \binom{15}{1} = 10 \cdot 15 = 150 $. The probability is $ \frac{150}{1140} = \frac{5}{38} $.

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