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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.

# Chinese Remainder Theorem

If a positive integer $$n$$ satisfies $\left\{\begin{matrix} n \equiv 2 & \pmod{3}\\ n \equiv 3 & \pmod {5}, \end{matrix}\right.$ how many possible values of $$n$$ are in the domain $$[10,29]?$$

If a positive integer $$n$$ satisfies $\left\{\begin{matrix} n \equiv 2 & \pmod{5} \\ n \equiv 3 & \pmod {9} \\ n \equiv 1 & \pmod {11}, \end{matrix}\right.$ what is the $$7 ^{\text{th}}$$ smallest possible value of $$n?$$

What is the largest positive integer $$n$$ satisfying $\left\{\begin{matrix} n \equiv 1 & \pmod{3}\\ n \equiv 2 & \pmod {7}, \end{matrix}\right.$ in the domain $$[0,600]?$$

If we make a sequence with positive integers $$n$$ satisfying $\left\{\begin{matrix} n \equiv 2 & \pmod{3}\\ n \equiv 3 & \pmod {4} \end{matrix}\right.$ in increasing order, what is the $$15 ^{\text{th}}$$ term?

If $$n \equiv 1 \pmod{3}$$ and $$n \equiv 3 \pmod { 7} ,$$ what is $$n?$$

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