Modular Arithmetic (Hard)

Modular Arithmetic: core idea

• 1. Smallest positive $n$ with $n\equiv 3\pmod8$, $n\equiv 4\pmod9$, $n\equiv 5\pmod7$.

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• 2. Last two digits of $7^{2024}$.

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• 3. $\varphi(720)$.

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• 4. Number of solutions to $x^2\equiv1\pmod{100}$.

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• 5. Least composite $n>1$ with $2^n\equiv2\pmod n$?

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• 6. $7^{100}\bmod 13$.

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• 7. How many $1\le n\le105$ are coprime to $105$?

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• 8. Order of $10$ modulo $21$.

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• 9. Inverse of $11$ modulo $26$.

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