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university of north carolina wilmington - Anadea

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 Factorize:

📌 $2025 = 3^4 \times 5^2$

📌 $1515 = 3 \times 5 \times 101$

📌 The $\gcd$ is the product of the smallest powers of common primes: $3^1 \times 5^1 = 15$.

📌 Thus, the maximum number of grants is $\boxed{15}$.

📌 Question: A spaceship’s navigation system uses a 13-bit code and a 17-bit engine control sequence. What is the smallest number of distinct sequences required to uniquely align both systems simultaneously?

📌 Solution: The smallest number of distinct sequences is the least common multiple (LCM) of 13 and 17.

📌 Since 13 and 17 are primes, $\text{LCM}(13, 17) = 13 \times 17 = 221$.

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Wait: let’s recompute at small $x$, say $x = 0.1$ radians:
$\sin x \approx 0.0998$, $\cos x \approx 0.995$, so $\sec x \approx 1....
$\sec x + \csc x \approx 11.025$, square $\approx 121.55$
$\sin x + \cos x \approx 1.0948$, square $\approx 1.20$
Sum $\approx 122.75$, which is much larger than 10.
So indeed, as $x \to 0^+$ or $x \to \frac{\pi}{2}^-$, the expression g...
But is that correct? Let's analyze:
As $x \to 0^+$, $\sec x \to 1$, $\csc x \to \infty$, so $(\sec x + \cs...
And $\sin x + \cos x \to 1$, so its square is constant, but the first...
Thus, $(\sec x + \csc x)^2 \to \infty$, so the entire expression tends...

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