🔢 number theory

1. **State the problem:** We want to determine if exact change can be made for the amounts 29, 19, 23, and 33 using an infinite supply of coins with denominations 6, 10, and 15. 2.

1. **State the problem:** Find the highest power of 20 that divides $50!$. 2. **Prime factorization of 20:**

1. **Problem statement:** We start with numbers 1, 2, 3, ..., 100 on the board. At each step, two numbers $a$ and $b$ are chosen and replaced by $|a - b|$. This continues until onl

1. **State the problem:** We want to determine which of the amounts 19, 29, 23, and 33 can be formed exactly using any number of coins of denominations 6, 10, and 15. 2. **Understa

1. **State the problem:** Find the highest power of 20 that divides 50!. 2. **Prime factorize 20:**

1. **State the problem:** We want to determine if exact change can be made for the amounts 23, 29, 19, and 33 using an infinite supply of coins with denominations 6, 10, and 15. 2.

1. **State the problem:** We want to prove by mathematical induction that the number $$U_n = 2^{6n} + 3^{2n-2}$$ is divisible by 5 for all positive integers $n$. 2. **Base case ($n

1. The problem asks if -9999 can be considered the smallest 4-digit number. 2. A 4-digit number is any integer from 1000 to 9999 or from -9999 to -1000 if considering negative numb

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