🔢 number theory

1. **Problem:** Find a positive integer $x$ such that $$x \equiv 2 \pmod{4}, \quad 2x \equiv 3 \pmod{9}, \quad 7x \equiv 1 \pmod{11}.$$

1. The problem is to understand the number 67. 2. 67 is a prime number, meaning it has no divisors other than 1 and itself.

1. The problem asks to identify composite numbers in the given sets. 2. Composite numbers are numbers greater than 1 that have more than two factors.

1. Masalani tushuntirish: 100 ta natural sonni $2a+3b$ ko'rinishida ifodalash mumkin emas, bunda $a$ va $b$ no manfiy butun sonlar (ya'ni $a,b \geq 0$) hisoblanadi. 2. Formulani ko

1. Muammo: 100 ta natural sonni $2a+3b$ ko'rinishida ifodalash mumkin emasligini aniqlash. 2. Formulalar va qoidalar: Bu yerda $a$ va $b$ butun sonlar, $a,b\geq 0$ deb olinadi. Biz

1. **State the problem:** Solve the congruence equation $$2x - 1 \equiv 2 \pmod{5}$$. 2. **Rewrite the equation:** Add 1 to both sides to isolate the term with $x$:

1. **State the problem:** We are given the number 378282 and need to understand or analyze it. 2. **Identify the nature of the number:** 378282 is a six-digit integer.

1. The problem is to understand or analyze the number 17626475831752227833399473284228. 2. This is a very large integer, and no specific operation or question is given.

1. The problem is to understand what a prime factor is. 2. A prime factor is a factor of a number that is a prime number itself.

1. **Problem statement:** Given $n$ consecutive integers $m, m+1, m+2, \ldots, m+n-1$, prove that there exists a non-empty subset of these integers whose sum is divisible by the su

1. **Problem Statement:** We are given a table with hexadecimal values and corresponding decimal values, and we need to find the formula or relation connecting them and predict the

1. **Stating the problem:** We want to test the theory of incrementing a large hexadecimal number by 16, four times, starting from the initial value \texttt{00000000000000000000000

1. **Problem Statement:** We need to prove that either $2 \cdot 10^{500} + 15$ or $2 \cdot 10^{500} + 16$ is not a perfect square. 2. **Key Idea:** Two consecutive integers cannot

1. **State the problem:** Calculate $254^{94} \bmod 160$ efficiently by hand. 2. **Simplify the base modulo 160:** Since $254 > 160$, reduce it first:

1. Асуудлыг тодорхойлно: 4-т хуваагдах бүх тоог сонгох. 2. 4-т хуваагдах тоо гэдэг нь 4-өөр хуваагдаж үлдэгдэл 0 гардаг тоо юм.

1. **Problem statement:** Given that $\gcd(a^n,b^n)=1$ and $\gcd(a,b^n)=1$, prove by induction that $\gcd(a^{n+1}+1,b^{n+1}+1)=1$. 2. **Base case (n=1):** We need to show $\gcd(a^{

1. **Problem statement:** Given that $\gcd(a^n,b^n) = 1$ and $\gcd(a,b^n) = 1$, show that $\gcd(a^{n+1}, b^{n+1}) = 1$. 2. **Recall the properties of gcd:**

1. لنفترض أن السؤال يتعلق بكيفية معرفة أن المتغير $d$ يقسم عددًا ما أو تعبيرًا رياضيًا. 2. في الرياضيات، نقول أن $d$ يقسم عددًا $n$ إذا كان هناك عدد صحيح $k$ بحيث أن $n = d \times

1. **State the problem:** We want to prove that if $\gcd(a,b) = 1$, then $\gcd(a+b, ab) = 1$. 2. **Recall the definition:** $\gcd(x,y)$ is the greatest positive integer that divide

1. **State the problem:** We want to prove that if $\gcd(a,b) = 1$, then $\gcd(a+b, ab) = 1$. 2. **Recall the definition:** $\gcd(x,y)$ is the greatest positive integer that divide

1. **Problem statement:** Given integers $a$, $b$, and $c$, none of which are zero simultaneously, and $d = \gcd(a,b,c)$, show that $$d = \gcd(\gcd(a,b), c) = \gcd(a, \gcd(b,c)) =