Number Theory

Basic Applications of Modular Arithmetic

Chinese Remainder Theorem

         

If a positive integer nn satisfies {n≡2(mod3)n≡3(mod5),\left\{\begin{matrix} n \equiv 2 & \pmod{3}\\ n \equiv 3 & \pmod {5}, \end{matrix}\right. how many possible values of nn are in the domain [10,29]?[10,29]?

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If a positive integer nn satisfies {n≡2(mod5)n≡3(mod9)n≡1(mod11),\left\{\begin{matrix} n \equiv 2 & \pmod{5} \\ n \equiv 3 & \pmod {9} \\ n \equiv 1 & \pmod {11}, \end{matrix}\right. what is the 7th7 ^{\text{th}} smallest possible value of n?n?

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What is the largest positive integer nn satisfying {n≡1(mod3)n≡2(mod7),\left\{\begin{matrix} n \equiv 1 & \pmod{3}\\ n \equiv 2 & \pmod {7}, \end{matrix}\right. in the domain [0,600]?[0,600]?

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If we make a sequence with positive integers nn satisfying {n≡2(mod3)n≡3(mod4)\left\{\begin{matrix} n \equiv 2 & \pmod{3}\\ n \equiv 3 & \pmod {4} \end{matrix}\right. in increasing order, what is the 15th15 ^{\text{th}} term?

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If n≡1(mod3) n \equiv 1 \pmod{3} and n≡3(mod7), n \equiv 3 \pmod { 7} , what is n?n?

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