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## Basic Applications of Modular Arithmetic

Solve integer equations, determine remainders of powers, and much more with the power of Modular Arithmetic.

# Level 2

The number of students in a school is a natural number and is between $$500$$ and $$600$$.

If we group them into groups of $$20$$, $$12$$ or $$36$$ each, $$7$$ students are always left over. How many students are in this school?

Find the remainder when $$2016!-2015!$$ is divided by $$2017$$.

You may use the fact that $$2017$$ is prime.

What is the remainder when $${7}^{88}$$ is divided by $$11?$$

What is the remainder when $$\Huge \color{red}{12}^{\color{green}{34}^{\color{blue}{56}^{\color{brown}{78}}}}$$ is divided by $$\color{indigo}{90}?$$

What is the smallest positive integer which is a multiple of 7, yet it gives a remainder of 1 when divided by any of 2, 3, 4, 5, or 6?

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