🔢 number theory

1. The problem asks for the smallest 4-digit number. 2. A 4-digit number is any number from 1000 to 9999.

1. The problem is to find the requirements for a four-digit number to be expressed as a product of two prime numbers. 2. Let the four-digit number be $N$, where $1000 \leq N \leq 9

1. समस्या: 100 और 1000 के बीच ऐसी संख्याएं खोजनी हैं जो 12 से भाग देने पर शेष 5 दें और 15 से भाग देने पर शेष 8 दें। 2. गणितीय रूप में, हमें $x$ ऐसी संख्या चाहिए जो निम्न शर्तें पूर

1. Statement of the problem. We are asked to compute $\gcd(2101,1009)$ and to solve the congruences $55x \equiv 34 \pmod{89}$ and $105x \equiv 143 \pmod{100}$.

1. The problem asks us to find integers $x$, $y$, and $z$ such that $$x^3 + y^3 + z^3 = 33.$$ 2. This is a famous type of Diophantine equation known as a sum of three cubes problem

1. **State the problem:** We need to show that $$-11100 \times 134 \equiv -1 \pmod{13}$$ without using a calculator. 2. **Reduce each number modulo 13:**

1. Problem: Given $m \cdot n = 24$ and $p_t$ is a prime number, find $p_t$ from options 38, 46, 73, 83. Since $p_t$ is prime and options are 38, 46, 73, 83, only 73 and 83 are prim

1. The problem asks us to verify the equality $2078 = 31303_5$, where the subscript 5 indicates that $31303_5$ is a number in base 5. 2. First, convert the base 5 number $31303_5$

1. נניח כי יש טבלה שבה עבור כל מספר ראשוני $p$ מהצורות $7 + 30n$, $11 + 30n$, $13 + 30n$, $29 + 30n$ בוחרים שני מספרים אקראיים $m_1, m_2$ בטווח $1$ עד $p$, כאשר $m_1 \neq m_2$. 2.

1. נניח כי יש טבלה המכסה את המוצר $$\prod_{i=0}^{n}\frac{5+30i}{7+30i}\frac{9+30i}{11+30i}\frac{11+30i}{13+30i}\frac{27+30i}{29+30i}$$

1. נניח שיש לנו טבלה עם אינדקסים $m,n$ ומספרים בטבלה הנתונים על ידי הנוסחה: $$a_k + c_k(m-1) + \bigl(b_k + 30(m-1)\bigr)(n-1)$$

1. Statement of the problem: Prove 6 is a factor of $n(n^2+5)$ for every integer $n$. 2. We will show the expression is divisible by 2 and by 3, and since 2 and 3 are coprime, it f

1. Let's start by understanding the problem: proving that a specific number is not rational means showing it cannot be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are i

1. The problem asks to find the smallest prime number $p$ such that the number $p^2 + p + 1$ is not prime. 2. We start testing prime numbers in increasing order:

1. **State the problem:** We want to prove that for all positive integers $n$, the product $n(n+1)(n+2)$ is divisible by 6. 2. **Understand divisibility by 6:** A number is divisib

1. We are given the equation $263 + 441 = 714$ and asked to determine the base in which this equation is true. 2. Let's represent the digits in base $b$ and convert them to decimal

1. Stating the problem: We want to find the smallest number by which 243 should be multiplied so that the product is a perfect cube. 2. Prime factorize 243:

1. We are asked to find the number of pairs of positive integers \(n\) and \(m\) such that \(1! + 2! + 3! + \cdots + n! = m^2\). 2. Let \(S_n = 1! + 2! + 3! + \cdots + n!\).

1. Problem: Explore rules for sums, products, and differences involving even and odd numbers. 2. Rule (i): Sum of two even numbers.

1. **State the problem:** We need to find all pairs of positive integers $(n,m)$ such that the sum of factorials from $1!$ to $n!$ equals a perfect square $m^2$, that is, $$1! + 2!

1. Nyatakan masalah: Cari nilai $x$ jika Faktor Sepunya Terbesar (FSTB) bagi 4, 8, dan $x$ ialah 2. 2. Ingat bahawa FSTB tiga nombor ialah faktor terbesar yang sama bagi ketiga-tig