🔢 number theory

1. Problema 4: Să demonstrăm că 24 divide expresia $$(5n^2 + 3)(n^4 + 8)$$ pentru orice număr natural $n$. 2. Observăm că 24 = 8 \times 3, deci trebuie să arătăm că expresia este d

1. **Enunțul problemei:** Avem un număr natural $n$ care, împărțit la $6$, $8$, $9$, $12$ și $16$, dă același rest $5$. 2. **Verificarea pentru $n=5$:** Împărțind $5$ la $6$, $8$,

1. **State the problem:** We want to prove that there are no integers $a$ and $b$ such that the equation $$12a - 28b = 11$$ holds. 2. **Recall the key concept:** For an equation of

1. **State the problem:** We want to prove that there are no integers $a$ and $b$ such that the equation $$12a - 13b = 11$$ holds. 2. **Recall the key concept:** For a linear Dioph

1. **Problem Statement:** We want to understand why, in proving there are infinitely many primes of the form $$4n+1$$, we use the number $$M=\left(2q_1q_2\cdots q_m\right)^2+1$$ in

1. **Problem Statement:** Prove that there are infinitely many primes of the form $4n+1$ and infinitely many primes of the form $4n-1$ without using modular arithmetic.

1. 問題の説明: \(\pi^{2^4}\) が超越数であることを証明します。\n\n2. 重要な定理の紹介: 超越数とは、任意の非零多項式の根でない複素数のことです。つまり、代数的でない数です。\n\n3. \(\pi\) は既知の超越数です。\n\n4. ルートの超越性に関する重要な結果として、\n\n - リンドマン・ヴァイエルシュトラスの定理に

1. **Problem statement:** We are given a positive integer $n$ with exactly 8 positive divisors, including 1 and $n$. Two of these divisors are 14 and 21. We need to find the sum of

1. The problem states the Riemann Hypothesis: For all complex numbers $\rho$, if $\zeta(\rho)=0$ and $0<\Re(\rho)<1$, then $\Re(\rho)=\frac{1}{2}$. 2. This is a famous conjecture i

1. **Problem Statement:** Prove that for every integer $a$ and positive integer $n$, the congruence $$a^{13} \equiv a \pmod{2730}$$

1. The problem asks to identify whether the number 16 is prime or composite. 2. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

1. The problem asks to identify the number described by the clues: Rational, Not an integer, Positive, Can be written as a terminating decimal. 2. A rational number can be expresse

1. The problem asks if 21 is a prime number. 2. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

1. Problem: Given the options 72, 54, 60, and 42, we need to identify the problem or find a relevant solution based on these numbers. 2. Since no explicit question is provided, let

1. **State the problem:** We are given a prime number $p$ such that $$p^2 \equiv 1.$$ We need to find which value of $p$ from the options satisfies this condition. 2. **Understand

1. The problem is to explain the number 7. 2. Seven is a natural number that comes after 6 and before 8.

1. **Stating the problem:** We want to find subsets of the digits of the number 141 that sum to the numbers 17, 32, 48, 60, and 64. 2. **Understanding subsets and sums:** A subset

1. **Stating the problem:** We need to use the numbers 1 through 9 to form a sum of 131, and from this sum, identify how to get the numbers 4, 16, 41, 48, and 66. 2. **Understandin

1. **Stating the problem:** We want to understand how to get the numbers 4, 16, 41, 48, and 66 from the number 131 using numbers from 01 through 69. 2. **Analyzing the problem:** T

1. **State the problem:** Prove that for any non-negative integer $n$, the expression $3^{2n} + 7$ is divisible by 8. 2. **Formula and approach:** We want to show that $3^{2n} + 7

1. **State the problem:** Prove that for any non-negative integer $n$, the expression $32n + 7$ is divisible by 8. 2. **Recall divisibility rules and formulas:** A number $a$ is di