🔢 number theory

1. **State the problem:** We need to find two prime numbers that add up to 30.

1. **Stating the problem:** We have three numbers with HCF (Highest Common Factor) 8 and LCM (Least Common Multiple) 2520.

1. **Problem statement:** Prove that for any positive integer $n$, the fraction $\frac{21n + 4}{14n + 3}$ is irreducible, meaning it cannot be simplified further. 2. **Key idea:**

1. **State the problem:** Find all positive integers $n$ such that $n^3 + 2n + 1$ is a perfect square. 2. **Set up the equation:** Let $k^2 = n^3 + 2n + 1$ where $k$ is an integer.

1. **Problem statement:** Prove that for any positive integers $m$ and $n$, the product $$(36m + n)(36n + m)$$ can never be a power of 2. 2. **Recall the definition:** A power of 2

1. Problem: Find the remainder when $7^{5284}$ is divided by 5. 2. Formula: Use modular arithmetic and Euler's theorem or Fermat's little theorem.

1. The problem is to create an equation using the digits 1 through 9 exactly once each, arranged to produce the highest possible integer value. 2. To maximize the integer value, we

1. **State the problem:** We need to find the greatest common divisor (gcd) $d$ of $a=135$ and $b=59$, and express $d$ as a linear combination $d = sa + tb$ where $s > 0$ is as sma

1. The problem asks us to identify which number among 4, 5, 9, and 15 is composite. 2. A composite number is a positive integer greater than 1 that has more than two distinct posit

1. **Problem 36:** Determine which of the given numbers divides $2^{15}$. 2. Recall that $2^{15} = 32768$ and is a power of 2, so its prime factorization is only 2's.

1. The problem asks us to classify the numbers 1, 6, 25, 29, 31, 33, 49, 51, and 53 into three categories: Prime, Composite, and Neither. 2. Definitions:

1. The problem asks for the prime number of 4. 2. Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.

1. **State the problem:** We need to classify the numbers in set A = { -6, \frac{1}{2}, -1.333... (3's repeat), \pi, 2, 5 } into (a) Natural numbers, (b) Integers, (c) Rational num

1. **Stating the problem:** We want to analyze the expression $7^n - 1$ where $n \in \mathbb{Z}^+$ (positive integers) and determine its divisibility by 6. 2. **Formula and rules:*

1. **Problem statement:** Find all positive integers $k>1$ such that there exists a positive integer $n$ with the property that $\binom{n}{k}$ is divisible by $n$, but for all $m$

1. **State the problem:** We are given the number 67 and need to understand or work with it as per the user's request. 2. **Since the user only provided the number 67 without a spe

1. **Problem Statement:** Show that the sum of a rational number and an irrational number is irrational. 2. **Definitions:**

1. The problem is to convert an integer from one base to another. 2. The general method involves two main steps: first, convert the number from the original base to base 10 (decima

1. **Problem statement:** Given a prime $p \in \mathbb{F}$ and the function

1. **Determine whether the number 57 is prime or composite.** A prime number has exactly two distinct positive divisors: 1 and itself.

1. The problem is to understand what an odd number is and to provide examples of odd numbers up to one hundred million. 2. An odd number is an integer which is not divisible by 2.