Divisibility problems

Author: SolveSheep

About modules, reminders and a lot of prime numbers




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1.5 Number Theory
Divisibility test

Determine whether \(2^{10} + 3^{5}\) is divisible by 7.
1.5 Algebra Number Theory
Modular Inverse Calculation (Prime Modulus)

Find the smallest positive integer \(x\) such that \(17x \equiv 1 \pmod{101}\).
2.0 Algebra Number Theory
Chinese Remainder Theorem with Prime Moduli

Find the smallest positive integer \(x\) that satisfies the following system of congruences:
\(x \equiv 3 \pmod 5\)
\(x \equiv 5 \pmod 7\)
\(x \equiv 7 \pmod{11}\)
2.0 Algebra Number Theory
Evaluating Large Powers Modulo a Prime

Calculate the remainder when \(5^{123456}\) is divided by \(17\).
2.0 Algebra Number Theory
Multiplicative Order Modulo a Prime

Find the smallest positive integer \(n\) such that \(3^n \equiv 1 \pmod{43}\).
2.0 Number Theory Combinatorics
Remainder of a Factorial Sum

Find the remainder when \(1! + 2! + 3! + \dots + 100!\) is divided by \(12\).
2.0 Number Theory
Remainder of a Large Power

Find the remainder when \(3^{2023}\) is divided by \(17\).
2.5 Number Theory
Prime Number Divisibility Condition

Find all prime numbers \(p\) such that \(p\) divides \(2^{p-1} + 1\).
2.5 Algebra Number Theory
Solving a Modular Power Congruence

Find the smallest positive integer \(x\) such that \(x^{10} \equiv 3 \pmod{11}\).
2.5 Algebra Number Theory
Sum of Powers Modulo a Prime

Let \(p\) be a prime number greater than 3. Evaluate the sum \(S = \sum_{k=1}^{p-1} k^{p-2} \pmod p\).
2.5 Number Theory
System of Modular Congruences

Find the smallest positive integer \(n\) that satisfies the following congruences:
\(n \equiv 2 \pmod{5}\)
\(n \equiv 3 \pmod{7}\)
\(n \equiv 4 \pmod{11}\).
3.0 Number Theory
Prime Divisor of Sum of Squares

Let \(p\) be a prime number. If \(p\) divides \(n^2 + 1\) for some integer \(n\), prove that \(p \equiv 1 \pmod{4}\) or \(p=2\).