🔢 number theory

1. The problem is to find the product of the first 12 prime numbers. 2. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.

1. **Problem statement:** We have $k$ distinct positive integers arranged in a circle such that the product of any two adjacent integers is less than 2026. We want to find the maxi

1. **Problem statement:** Find the sum of all positive integers $n \leq 200$ such that the sum of all distinct positive divisors of $n$ equals $2n - 1$. 2. **Understanding the prob

1. **Stating the problem:** We want to prove that there exist both rational and irrational numbers. 2. **Definition of rational numbers:** A rational number is any number that can

1. نبدأ بقراءة المسألة: لدينا عدد طبيعي $N$ مكون من رقمين. 2. المعطيات:

1. **Problem Statement:** We are given the recursive definition of Euclidean numbers: $$e_1 = 2$$

1. **Problem:** Solve the linear Diophantine equation $$172x + 20y = 1000$$ completely. 2. **Step 1: Find the gcd of 172 and 20.**

1. **State the problem:** Solve the linear Diophantine equation $$172x + 20y = 1000$$ for integers $x$ and $y$. 2. **Recall the condition for solutions:** A linear Diophantine equa

1. The problem is to solve for the value of number 19, which is ambiguous as stated. Assuming you want to understand the number 19 or solve a problem involving 19, please clarify.

1. **Problem Statement:** Investigate the last digits of integers raised to powers, specifically focusing on the pattern of the last one or two digits. 2. **Understanding the Probl

1. 問題陳述： 已知A同B都係21600嘅正因數，且A>B。求有幾多種選擇A同B嘅方法，使得兩個數嘅最大公因數（GCD）係1。

1. **Problem:** Determine if the numbers 51, 87, and 59 are prime or not prime. 2. **Recall:** A prime number is a number greater than 1 that has no positive divisors other than 1

1. The user asks: "what is the name of the proof for 3". 2. This question is ambiguous because "proof for 3" is not a standard mathematical phrase.

1. **Problem statement:** For each $n = 84$ and $n = 88$, find the smallest integer multiple of $n$ whose base 10 representation consists entirely of digits 6 and 7. 2. **Approach:

1. **Problem Statement:** We have three types of chocolate bags containing 6, 9, and 20 chocolates respectively. We want to find the largest number of chocolates that cannot be obt

1. Problem: Determine the remainder when $2026^{2026}$ is divided by 17. 2. Formula and rules: We use modular arithmetic and Fermat's Little Theorem which states that for a prime $

1. Problem: Does 17 divide each of these numbers? a) 68 b) 84 c) 357 d) 1001 2. To check if 17 divides a number $n$, we verify if $n \mod 17 = 0$.

1. **Problem:** Does 17 divide each of these numbers? a) 68 b) 84 c) 357 d) 1001 2. **Formula and rule:** An integer $a$ divides another integer $b$ (written $a \mid b$) if there e

1. **State the problem:** Find the exponent of 3 in the prime factorization of 2025. 2. **Recall the prime factorization process:** To find the exponent of a prime in a number, rep

1. **Problem statement:** We have a positive integer sequence $(t_n)_{n\geq 1}$ defined by the recurrence relation $$t_{n+2} = \frac{t_n + t_{n+1}}{\gcd(t_n, t_{n+1})}$$

1. The Riemann Hypothesis is a famous unsolved problem in mathematics concerning the zeros of the Riemann zeta function $\zeta(s)$.\n\n2. The hypothesis states that all non-trivial