Number Theory

Modular Arithmetic Operations

Modular Arithmetic Operations: Level 2 Challenges

         

Find the last digit when

11+22+33++99+1010\color{#3D99F6}1^{\color{#3D99F6}1} + \color{#20A900}2^{\color{#20A900}2} + \color{#69047E}3^{ \color{#69047E}3} + \cdots + \color{#D61F06}9^{ \color{#D61F06}9} + \color{#BA33D6}{10}^{\color{#BA33D6}{10}}

is written out as an integer.

What is 51(mod17)? 5 ^ {-1} \pmod{17} ?

Hint: Remember that inverses multiply to 1.

Given that 2292^{29} is a nine-digit number with distinct digits, determine (without evaluating 2292^{29}) which one of the ten digits is missing.

What is the sum of all possible primes pp such that p2+8p^2+8 is also a prime?

How many natural numbers nn exist such that the following are all primes?

3n44n55n33n-4 \qquad 4n-5 \qquad 5n-3

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