🔢 number theory

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

1. **Problem statement:** Find the number of pairs of consecutive integers in the set $\{1026, 1027, 1028, \ldots, 2026\}$ such that when these two integers are added, no carrying

1. **Problem statement:** Find the number of pairs of consecutive integers in the set $\{2026, 2027, 2028, \ldots, 3026\}$ such that no carrying is required when adding the two int

1. **Stating the problem:** We want to find the highest possible number of books you have, which is less than 75, and this number must be a prime number (divisible only by 1 and it

1. The problem is to find the last non-zero digit of a very large number calculated by Niloy. 2. Usually, such problems involve factorials or large products where trailing zeros ap

1. **Problem statement:** We need to find how many times the digit 4 appears in the page numbers from 1 to 486. 2. **Approach:** We will count the occurrences of digit 4 in each di

1. The problem asks about the name given to certain primes. 2. In mathematics, primes that satisfy specific properties often have special names.

1. **Problem statement:** Find all prime numbers $p$ such that $p-3$ is a perfect square. 2. **Set up the equation:** Let $p-3 = n^2$ where $n$ is an integer.