🔢 number theory

1. **Find the number of positive integers \( \leq 3000 \) divisible by 3, 5 or 7.** We use the Inclusion-Exclusion Principle.

1. **Stating the problem:** Classify the numbers 2.7, \(\frac{2}{4}\), and 2 \(\frac{8}{9}\) into the sets: Real Numbers, Irrational Numbers, Rational Numbers, Integers, Whole Numb

1. **State the problem:** We need to find the value of $d$ such that $5d \equiv 1 \pmod{96}$. This means $5d$ leaves a remainder of 1 when divided by 96. 2. **Formula and concept:*

1. The problem is to find the sequence using Euler's theorem. 2. Euler's theorem states that if $a$ and $n$ are coprime (i.e., $\gcd(a,n)=1$), then:

1. **Stating the problem:** We want to identify prime numbers easily. 2. **Definition:** A prime number is a natural number greater than 1 that has no positive divisors other than

1. **Problem Statement:** We need to find the multiplicative inverse of $a=5$ modulo a prime number $p=17$. This means finding an integer $x$ such that: $$5 \times x \equiv 1 \pmod

1. Problem: Explain why $-(m+1)!(p-m-2)!\equiv(-1)^{m+1}\pmod{p}$ becomes $(m+1)!(p-m-2)!\equiv(-1)^{m+2}\pmod{p}$ and prove that for a prime $p$ and $0\le k\le p-1$ we have $k!(p-

1. **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. **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