Number Theory
# Euler's Theorem

What is the last digit of \( 2^9?\)

For a positive integer \(n>2,\) what can we say about the number of positive integers less than \(n\) that are relatively prime to \(n\) (that is, their GCD is 1)?

Hint: A useful property of the Greatest Common Divisor function is \(\gcd(a,b) = \gcd(a,a-b);\) e.g., \(\gcd(10,8) = \gcd(10,2) = 2.\)

Find the last two digits of \(7^{100}-3^{100}.\)

What is the largest integer that always divides \(n^{7}-n\) for any integer \(n?\)

\[\LARGE \color{blue}5^{\color{green}5^{\color{red}5^{\color{brown}5^\color{magenta}5}}}\]

What are the last three digits of the number above?

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