🔢 number theory

1. **Problem statement:** Find the number of pairs of consecutive integers in the set $\{1026, 1027, 1028, \ldots, 2026\}$ such that when these two integers are added, no carrying

1. **Problem statement:** Find the number of pairs of consecutive integers in the set $\{2026, 2027, 2028, \ldots, 3026\}$ such that no carrying is required when adding the two int

1. **Stating the problem:** We want to find the highest possible number of books you have, which is less than 75, and this number must be a prime number (divisible only by 1 and it

1. The problem is to find the last non-zero digit of a very large number calculated by Niloy. 2. Usually, such problems involve factorials or large products where trailing zeros ap

1. **Problem statement:** We need to find how many times the digit 4 appears in the page numbers from 1 to 486. 2. **Approach:** We will count the occurrences of digit 4 in each di

1. The problem asks about the name given to certain primes. 2. In mathematics, primes that satisfy specific properties often have special names.

1. **Problem statement:** Find all prime numbers $p$ such that $p-3$ is a perfect square. 2. **Set up the equation:** Let $p-3 = n^2$ where $n$ is an integer.

1. The problem is to understand the number 67. 2. 67 is a positive integer.

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