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Modular Arithmetic Operations

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

Level 2

         

Find the last digit when \[\large \color{blue}1^{\color{blue}1} + \color{green}2^{\color{green}2} + \color{purple}3^{ \color{purple}3} + \ldots + \color{red}9^{ \color{red}9} + \color{violet}{10}^{\color{violet}{10}} \] is written out as an integer.

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

Hint: Remember that inverses multiply to 1.

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.

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

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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