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Number Theory

Number Theory Warmups

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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