🔢 number theory

1. **Problem:** Express the decimal number $X=380$ in bases 2, 3, 7, 8, and 16. 2. **Formula and rules:** To convert a decimal number to another base $b$, repeatedly divide the num

1. **State the problem:** Solve the system of linear congruences. Since the user did not specify the system, let's consider a general example: $$\begin{cases} x \equiv a_1 \pmod{m_

1. **State the problem:** We need to show that 43 is a prime number and find the prime factorization of 2560. 2. **Show that 43 is prime:** A prime number is a number greater than

1. Bài toán: Cho $p$ và $q$ là số nguyên tố, và $m^{p-1} \equiv 1 \pmod{p}$, $m^{q-1} \equiv 1 \pmod{q}$. Hỏi có thể nhân hai biểu thức đồng dư này với nhau được không? 2. Công thứ

1. The problem is to find another 6-digit number that exhibits the same cyclic behavior as 142857 when multiplied or added repeatedly. 2. The number 142857 is the cyclic number gen

1. The problem is to understand why adding the number 142857 repeatedly results in the same digits repeating up to 6 times. 2. The number 142857 is a special cyclic number related

1. **State the problem:** We want to find the remainder when $289^{125}$ is divided by 27, i.e., compute $289^{125} \bmod 27$. 2. **Simplify the base modulo 27:** Since $289$ is la

1. The problem is to find the remainder when $289^{125}$ is divided by 27, i.e., compute $289^{125} \bmod 27$. 2. We use modular arithmetic properties and Euler's theorem or simpli

1. The problem is to find the remainder when $289^{125}$ is divided by 19, i.e., compute $289^{125} \bmod 19$. 2. We use modular arithmetic properties and Fermat's Little Theorem w

1. **Problem statement:** We are given a number $n$ such that when its square root is divided by 11, the remainder is 6, and $6 < \sqrt{n} < 28$. We need to find $n$ and then compu

1. **Problem statement:** We have two numbers: $(ab)_4$ in base 4 and $(ba)_7$ in base 7, where $a$ and $b$ are digits. We want to find the largest possible value of $(a+b)_{10}$ a

1. **Problem statement:** We are given two numbers: $(ab)_4$ in base 4 and $(ba)_7$ in base 7, where $a$ and $b$ are digits. We want to find the largest possible value of $(a+b)_{1

1. **State the problem:** We want to find how many 9-digit numbers are less than 3000 and divisible by 9. 2. **Analyze the problem:** A 9-digit number is any number from 100000000

1. **Problem statement:** We are given a number 85AB1 which is a multiple of 99. We need to find the digits $A$ and $B$. 2. **Key fact:** A number is divisible by 99 if and only if

1. مسئله: محاسبه باقیمانده تقسیم $$(-6)^{23}$$ بر 33. 2. فرمول و قانون مهم: برای محاسبه باقیمانده تقسیم توان‌های بزرگ، از خواص حساب باقی‌مانده (حلقه‌های مدولار) استفاده می‌کنیم.

1. **Problem Statement:** Find all palindrome numbers less than 100, determine if there is a common factor for all these palindrome numbers, and classify these numbers as rational

1. The problem asks: How many numbers from 50 to 500 have an odd number of factors? 2. Important fact: A number has an odd number of factors if and only if it is a perfect square.

1. **Problem statement:** We need to find all positive prime numbers $p$ such that $2p+1$ is a perfect cube. 2. **Formula and approach:** Let $2p+1 = n^3$ for some integer $n$. Sin

1. The problem asks: How many numbers from 50 to 500 have an odd number of factors? 2. Important fact: A number has an odd number of factors if and only if it is a perfect square.

1. **Problem statement:** We need to find three distinct prime numbers whose sum is 30 and whose product is as large as possible. 2. **Recall prime numbers:** Prime numbers are num

1. সমস্যাটি হলো: সেট {2026, 2027, 2028, \cdots, 3026} থেকে এমন ক্রমিক পূর্ণসংখ্যার জোড়া খুঁজে বের করতে হবে যাদের যোগ করার সময় কোন হাতের স্থানান্তর (carrying) হয় না। 2. সূত্র ও ন