Number Theory

Modular Arithmetic Operations

Modular Arithmetic Operations: Level 3 Challenges

         

What is the remainder when 12013+22013++20122013+201320131^{2013}+2^{2013}+\cdots +2012^{2013}+2013^{2013} is divided by 20142014?

When we rotate an integer, we take the last digit (right most) and move it to the front of the number. For example, if we rotate 1234512345, we will get 51234 51234 .

What is the smallest (positive) integer NN, such that when NN is rotated, we obtain 23N \frac{2}{3} N ?

Find the remainder when 698+8986^{98}+8^{98} is divided by 98.

Find the smallest positive integer kk such that 12+22+32++k21^2+2^2+3^2+\ldots+k^2 is a multiple of 200.

axa2(mod(a1)){a^x \equiv a-2 \pmod{{\small(a-1)}}}

If aa and xx are positive integers greater than 2, what is the value of a?a?

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