🔢 number theory

1. **State the problem:** We want to find the probability that $2^m + 3^n$ is divisible by $35$ when $m$ and $n$ are chosen from $\{1, 2, \dots, 600\}$ independently and uniformly.

1. Bài toán yêu cầu tìm số dư của $2004^{376}$ khi chia cho $1975$. 2. Ta sử dụng định lý Fermat hoặc định lý Euler để tính số dư của lũy thừa lớn modulo một số.

1. Bài toán yêu cầu tìm số dư của $2004^{376}$ khi chia cho $1975$. Đây là bài toán về tính toán số dư của lũy thừa lớn modulo một số. 2. Công thức sử dụng: Để tính $a^b \bmod m$,

1. The problem is to explain the number $6$ in more detail. 2. The number $6$ is a natural number that comes after $5$ and before $7$.

1. **Nyatakan masalah:** Kita diberi nombor dalam asas enam, iaitu $3412_6$, dan diberikan pengembangan $3412_6 = 3 \times 6^3 + 4 \times 6^m + n + 2 \times 6^0$. Kita perlu cari n

1. Задача: Найти все натуральные числа $n$, для которых сумма цифр числа $5^n$ равна $2n$. 2. Формула и правила: Сумма цифр числа — это сумма всех его десятичных цифр. Нужно провер

1. **State the problem:** We want to find the exact total number of times Ali stabbed the 2020 people following the rule that at the $i$-th time, he stabs every person whose number

1. **State the problem:** Ali stabs $2020$ people arranged in a circle. On the $i$-th round, he stabs every person whose number is a multiple of $i$ exactly once. We want to find t

1. **Problem statement:** There are 2026 integers greater than 1 on a blackboard. In each move, two integers $m>1$ and $n>1$ are chosen from different places and replaced by $\gcd(

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