Bezout’s Identity

Bezout’s Identity states that for any natural numbers $a$ and $b$, there exist integers $x$ and $y$, such that $$ \text{gcd}(a, b) = ax + by $$ Problem (42 Points Training, 2018) Let $p$ be a prime, $p>2$. Prove that any prime divisor of the number $2^p-1$ has the form…

Mathematical Induction: Inequalities

Mathematical Induction is used to prove that the statement of the problem $P(n)$ is true for all natural numbers $n$. Mathematical Induction consists of proving the following three theorems. Theorem 1 (Base of Induction): The statement of the problem is true for $n = 1$. Theorem 2 (Inductive Step): If…

Mathematical Induction: Combinatorics

Mathematical Induction is used to prove that the statement of the problem $P(n)$ is true for all natural numbers $n$. Mathematical Induction consists of proving the following three theorems. Theorem 1 (Base of Induction): The statement of the problem is true for $n = 1$. Theorem 2 (Inductive Step): If…

Mathematical Induction: Algebra

Mathematical Induction is used to prove that the statement of the problem is true for all natural numbers $n$. Mathematical Induction consists of proving the following three theorems. Theorem 1 (Base of Induction): The statement of the problem is true for $n = 1$. Theorem 2 (Inductive Step): If the…

Mathematical Induction: Number Theory

Mathematical Induction is used to prove that the statement of the problem is true for all natural numbers $n$. Mathematical Induction consists of proving the following three theorems. Theorem 1 (Base of Induction): The statement of the problem is true for $n = 1$. Theorem 2 (Inductive Step): If the…