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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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