Number Theory
# Number Theory Warmups

One of the seven goblets above is made of real gold. If you start counting at A and wind back and forth while counting (A, B, C, D, E, F, G, F, E, D, ...), then the golden goblet would be the \(1000^\text{th}\) one that you count.

Which one is the golden goblet?

\[\Huge {\color{blue}9}^{{\color{green}8}^{{\color{red}7}^{{\color{brown}6} ^{\color{magenta}5}}}}\]

What are the last two digits when this integer fully expanded out?

There is a prime number \(p\) such that \(16p+1\) is the cube of a positive integer. Find \(p\).

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