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Number Theory Warmups

If numbers aren't beautiful, we don't know what is. Dive into this fun collection to play with numbers like never before, and start unlocking the connections that are the foundation of Number Theory.

Number Theory Warmups: Level 3 Challenges


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?

Find the sum of all positive integers \(\displaystyle n\), such that \(\displaystyle \dfrac{(n+1)^2}{n+7}\) is an integer.

Find the sum of all prime numbers \(p\) such that \(p|\underset { p }{ \underbrace { 111\dots 1 } } \).

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


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