🔢 number theory

1. **Problem Statement:** Determine the largest positive integer $n$ for which there exist positive integers $a$ and $q$ such that

1. مسئله: با استفاده از روش غربال اراتستن، اولین عددی که در دسته‌ی مضارب ۱۳ خط می‌خورد را پیدا کنید. 2. روش غربال اراتستن برای یافتن اعداد اول به این صورت است که ابتدا اعداد طبیعی

1. The problem is to understand what "6 chiffres" means in a mathematical context. "Chiffres" is French for "digits". 2. If the question is about the number of 6-digit numbers, we

1. Мәселені айқындау: Бізге $p - q + r = \sqrt{p + q + r}$ теңдеуін қанағаттандыратын жай сандардың $(p, q, r)$ үштіктерін табу керек. 2. Теңдеуді қарастырайық: $p - q + r = \sqrt{

1. **Problem Statement:** Given that $p$ and $q$ are both prime numbers, we want to understand what this implies and explore some properties. 2. **Definition of Prime Numbers:** A

1. **State the problem:** We are given a system of congruences: $$3x \equiv 5 \pmod{19},$$

1. **Stating the problem:** We are given a function $$K = n \ln n + \ln \ln n - n + 2.25 \left(\frac{\ln n!}{\ln n}\right) + \frac{n^s}{\ln n}$$ and several values of $n$ with corr

1. **Problem:** There are between 50 and 60 eggs in a basket. When counted by 3's, remainder is 2. When counted by 5's, remainder is 4. Find the number of eggs. 2. **Formula and ru

1. The problem asks us to identify which factors of 70 are not prime. 2. Recall that a prime number is a number greater than 1 that has no positive divisors other than 1 and itself

1. The problem asks us to identify which factors of 70 are not prime. 2. Recall that a prime number is a number greater than 1 that has no positive divisors other than 1 and itself

1. The problem is to find numbers by which you can divide two or three given numbers. 2. To solve this, we use the concept of the Greatest Common Divisor (GCD), which is the larges

1. The problem is to find a number that can divide all the given numbers (3845966 and any others implied). 2. This is a problem of finding the Greatest Common Divisor (GCD) or High

1. **Stating the problem:** We have a list of numbers with decimals and want to decompose each into whole numbers by dividing by a divisor with 3 to 4 digits (i.e., between 100 and

1. **Problem statement:** Find the least number which when divided by 24 and 32 leaves a remainder of 6 in each case. 2. **Understanding the problem:** If a number $N$ leaves a rem

1. Problem: Find all solutions for the system of linear congruences using the Chinese Remainder Theorem (CRT) or substitution method as specified. 2. For part (a):

1. **State the problem:** Solve the system of congruences: $$x \equiv 1 \pmod{3}, \quad x \equiv 2 \pmod{5}, \quad x \equiv 3 \pmod{7}$$

1. Мәселені түсіндіру: Ең кіші ортақ еселігі (ЕКОЕ) 15 болатын натурал сандар жұптарының санын табу керек. 2. ЕКОЕ анықтамасы: Екі санның ең кіші ортақ еселігі — бұл екі санның еке

1. **State the problem:** Solve the system of congruences: $$3x + 13y \equiv 8 \pmod{55}$$

1. **Problem Statement:** We are given a positive integer $n$ and three statements about divisibility involving $n$ and $n^2$. We need to determine which statements are true. 2. **

1. समस्या: वह सबसे छोटी संख्या $x$ ज्ञात करें जो 12, 16, 18, 20 और 25 से विभाजित करने पर प्रत्येक बार शेष 4 देती है, और $x$ 7 से विभाजित होती है। 2. सूत्र और नियम: यदि कोई संख्या $

1. **Problem Statement:** Prove that among any 16 consecutive integers, there is at least one integer that is relatively prime (shares no common divisor greater than 1) to each of