Mathematics & Logic

Integer Labyrinth: Unraveling Prime and Modular Mysteries

Each problem in this series demands a sharp grasp of advanced number‑theoretic tools to navigate the subtle patterns hidden within ordinary integers.

#Fermat#quadratic residues#Diophantine#congruences#integer properties#primality#modular inverses#Euler's theorem

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Total Questions

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hard

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Quiz Questions

Answer all questions below and test your knowledge.

1
Let p be an odd prime. For how many integers a, with 1 ≤ a ≤ p−1, does the congruence a^(p−1) ≡ 1 (mod p²) hold?
Question 1
2
If n is a positive integer such that φ(n) divides n−1, where φ is Euler's totient function, what can be concluded about n?
Question 2
3
Let k be the smallest positive integer such that 2^k ≡ 1 (mod 25). What is the remainder when k is divided by 5?
Question 3
4
How many integer solutions (x, y) are there to the equation x^2 + 2xy + 2y^2 = 2026?
Question 4
5
Let p be a prime greater than 3. What is the remainder when the product of all quadratic residues modulo p (excluding 0) is divided by p?
Question 5
6
How many positive integers n ≤ 2026 satisfy that the sum of the positive divisors of n (σ(n)) is odd?
Question 6
7
Let m and n be positive integers such that m² + n² is divisible by 49. What is the minimum possible value of m+n?
Question 7
8
What is the largest integer n such that 10^n divides 2026! ?
Question 8

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Integer Labyrinth: Unraveling Prime and Modular Mysteries

Quiz Intelligence

Quiz Questions

Let p be an odd prime. For how many integers a, with 1 ≤ a ≤ p−1, does the congruence a^(p−1) ≡ 1 (mod p²) hold?

If n is a positive integer such that φ(n) divides n−1, where φ is Euler's totient function, what can be concluded about n?

Let k be the smallest positive integer such that 2^k ≡ 1 (mod 25). What is the remainder when k is divided by 5?

How many integer solutions (x, y) are there to the equation x^2 + 2xy + 2y^2 = 2026?

Let p be a prime greater than 3. What is the remainder when the product of all quadratic residues modulo p (excluding 0) is divided by p?

How many positive integers n ≤ 2026 satisfy that the sum of the positive divisors of n (σ(n)) is odd?

Let m and n be positive integers such that m² + n² is divisible by 49. What is the minimum possible value of m+n?

What is the largest integer n such that 10^n divides 2026! ?

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