🔢 number theory

1. **Problem Statement:** Prove that there are infinitely many even numbers. 2. **Definition:** An even number is any integer that can be written as $2k$ where $k$ is an integer.

1. **Problem Statement:** Define the multiplicative inverse in modular arithmetic and find the multiplicative inverse of $5$ modulo $9$.

1. **Problem:** Find two pairs of reversed prime numbers. **Step 1:** Understand reversed primes are pairs like $13$ and $31$ where digits are reversed and both numbers are prime.

1. **State the problem:** We need to find Pablo's and Sebastian's ages based on the sum of the prime factors of 99 and 88 respectively.

1. **Stating the problem:** We want to understand why the number $\pi$ is irrational. 2. **Definition:** A number is irrational if it cannot be expressed as a fraction $\frac{a}{b}

1. The problem states: "A prime number has exactly two factors: 1 and itself." This is a definition, not a problem to solve. 2. Important rule: A prime number is a natural number g

1. **Problem Statement:** We need to find two prime numbers that add up to a given even number.

1. **Problem Statement:** Define the multiplicative inverse in modular arithmetic and find the multiplicative inverse of $5$ modulo $9$.

1. **Problem Statement:** We want to understand why for any polynomial $f(x)$ and prime $p$, every solution to the congruence

1. **Stating the problem:** We have a function defined for any positive integer $n$ as $$J(n) := n \times \prod_{p|n} \frac{1 + p}{p}$$ where the product is over all prime divisors

1. **State the problem:** We need to determine if Thomas's claim that all square numbers (except 0) have an odd number of factors is correct. 2. **Recall the rule about factors:**

1. **Problem Statement:** We want to determine if the set $\{f_n \sqrt{2} \bmod 1\}$ is dense in the interval $[0,1]$, where $f_n$ is a sequence of integers. 2. **Key Concept:** A

1. **Problem Statement:** We want to determine if the set $\{f_n \sqrt{2} \bmod 1\}$, where $f_n$ is the $n$th Fibonacci number, is dense in the interval $[0,1]$.

1. The problem asks us to identify which numbers from the list are prime numbers. 2. A prime number is a natural number greater than 1 that has no positive divisors other than 1 an

1. **State the problem:** We have a first column of numbers: $28$, $42$, and $14$. Adding their digits gives $2 + 8 + 4 + 2 + 1 + 4 = 21$. Then adding the digits of $21$ gives $2 +

1. The user asked about the number $2024$. 2. Since no specific operation or context was given, let's analyze $2024$ as a number.

1. The numbers $67$ and $61$ are both integers. 2. They are prime numbers, meaning they have no divisors other than $1$ and themselves.

1. The problem presents a list of numbers each followed by a number in parentheses, e.g., $3435 (6)$, $124 (7)$, etc. 2. We need to understand the relationship between the numbers

1. **بيان المسألة:** نريد تحديد جميع الأعداد الزوجية الموجبة $n$ التي تحقق الشرط التالي:

1. **بيان المسألة:** نريد تحديد جميع الأعداد الطبيعية الموجبة $n>1$ بحيث إذا كانت $d_1, d_2, \ldots, d_k$ هي كل القواسم الموجبة للعدد $n$ مرتبة تصاعدياً بحيث $1=d_1 < d_2 < \ldots

1. **State the problem:** We want to find the probability that $2^m + 3^n$ is divisible by $35$ when $m$ and $n$ are chosen from $\{1,2,\dots,600\}$ independently and uniformly. 2.