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 that divides 1111111111111111the digit 1 is repeated p times\underset{\text{the digit 1 is repeated }p \text{ times}}{\underbrace{111111111111111\cdots 1}}.

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


Problem Loading...

Note Loading...

Set Loading...