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Considering the remainder "modulo" an integer is a powerful, foundational tool in Number Theory. You already use in clocks and work modulo 12.

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What is \( 5 ^ {-1} \pmod{17} ?\)

**Hint:** Remember that inverses multiply to 1.

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Given that \(2^{29}\) is a nine-digit number with distinct digits, determine (without evaluating \(2^{29}\)) which one of the ten digits is missing.

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What is the sum of all possible primes \(p\) such that \(p^2+8\) is also a prime?

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How many natural numbers \(n\) exist such that the following are all primes?

\[3n-4 \qquad 4n-5 \qquad 5n-3\]

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