Exercises - Problem Sets 1

Here are 12 practice problems on modulo arithmetic, ranging from simple calculations to more challenging concepts:

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Practice Problems: Modulo Arithmetic

Problem 1. Find the remainder:

\( 37 \mod 5 \)

Problem 2. Simplify using modulo:

\( (25 + 13) \mod 7 \)

Problem 3. Solve for \( x \):

\( x \equiv 5 \mod 9 \)
\( x + 17 \equiv ? \mod 9 \)

Problem 4. Modular exponentiation:

\( 3^4 \mod 11 \)

Problem 5. Congruence equation:

Solve for \( x \):
\( 7x \equiv 3 \mod 10 \)

Problem 6. Chinese Remainder Theorem:

Solve the system:
[ x \equiv 2 \mod 5
x \equiv 3 \mod 7 ]

Problem 7. Multiplicative inverse:

Find the multiplicative inverse of \( 4 \mod 9 \).

Problem 8. Simplify a product modulo:

\( (14 \times 21) \mod 10 \)

Problem 9. Quadratic congruence:

Solve for \( x \) in:
\( x^2 \equiv 4 \mod 9 \)

Problem 10. Large modulus calculation:

\( 123456 \mod 13 \)

Problem 11. Euler’s Theorem application:

Find \( 5^{100} \mod 14 \) using Euler’s theorem.

Problem 12. Fermat’s Little Theorem:

Use Fermat's Little Theorem to find \( 7^{51} \mod 11 \).

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These problems range from basic calculations to more advanced applications like modular inverses, the Chinese Remainder Theorem, and number theory concepts like Euler's and Fermat's theorems.

Click here for the solutions.