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 1000th1000^\text{th} one that you count.

Which one is the golden goblet?

98765\Huge {\color{#3D99F6}9}^{{\color{#20A900}8}^{{\color{#D61F06}7}^{{\color{#624F41}6} ^{\color{magenta}5}}}}

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

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

Find the sum of all prime numbers pp such that p1111pp|\underset { p }{ \underbrace { 111\dots 1 } } .

There is a prime number pp such that 16p+116p+1 is the cube of a positive integer. Find pp.

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