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Home » B: Faradayâs Law of Electromagnetic Induction - Groen Casting

B: Faradayâs Law of Electromagnetic Induction - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Solution: Let the number be $n$. Then $n \equiv 3 \pmod{7}$ and $n \equiv 5 \pmod{6}$. Solve the system:

📌 From $n \equiv 5 \pmod{6}$, $n = 6k + 5$. Substitute into the first congruence: $6k + 5 \equiv 3 \pmod{7} \Rightarrow 6k \equiv -2 \pmod{7} \Rightarrow 6k \equiv 5 \pmod{7}$. Multiply both sides by the modular inverse of 6 mod 7 (which is 6, since $6 \times 6 = 36 \equiv 1 \pmod{7}$): $k \equiv 5 \times 6 \equiv 30 \equiv 2 \pmod{7}$. Thus, $k = 7m + 2$, and $n = 6(7m + 2) + 5 = 42m + 17$. The two-digit solutions are when $m = 0$: $n = 17$ and $m = 1$: $n = 59$. The smallest is $\boxed{17}$.

📌 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}}

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\frac{4}{3}\pi r^3 - \frac{4}{3}\pi (r - 2)^3 = 128\pi
Divide through by \( \frac{4}{3}\pi \):
r^3 - (r - 2)^3 = 96
Expand \( (r - 2)^3 \):
r^3 - (r^3 - 6r^2 + 12r - 8) = 96
6r^2 - 12r + 8 = 96
6r^2 - 12r - 88 = 0
3r^2 - 6r - 44 = 0
r = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot (-44)}}{2 \cdot 3} = \f...
\sqrt{564} = \sqrt{4 \cdot 141} = 2\sqrt{141}

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