🔢 number theory

1. The problem is to understand what an even number is and to see examples of even numbers up to one hundred million. 2. An even number is any integer that can be exactly divided b

1. The problem asks us to identify which numbers among 15, 13, 9, and 8 are prime. 2. A prime number is a natural number greater than 1 that has no positive divisors other than 1 a

1. The problem is to identify which numbers among 49, 47, 13, and 21 are prime. 2. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and

1. **State the problem:** Solve the congruence equation $$13x \equiv 71 \pmod{380}$$. 2. **Formula and rules:** To solve $$ax \equiv b \pmod{m}$$, we need to find the modular inver

1. **State the problem:** Find the greatest number that divides 43, 91, and 183 leaving the same remainder in each case. 2. **Key idea:** If a number $d$ divides these numbers leav

1. **State the problem:** We need to show that every term of the sequence defined by $U_k = 2(4^k) + 1$ is divisible by 3 for all integers $k \geq 0$. 2. **Recall the divisibility

1. সমস্যাটি হলো: ১০০৮ এর মোট ভাজক (divisors) কতটি আছে তা নির্ণয় করা। 2. ভাজক নির্ণয়ের সূত্র: একটি সংখ্যার মৌলিক গুণনীয়করণ করলে যদি $$n = p_1^{a_1} \times p_2^{a_2} \times \cdots

1. The problem is to find the product of the first 12 prime numbers. 2. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.

1. **Problem statement:** We have $k$ distinct positive integers arranged in a circle such that the product of any two adjacent integers is less than 2026. We want to find the maxi

1. **Problem statement:** Find the sum of all positive integers $n \leq 200$ such that the sum of all distinct positive divisors of $n$ equals $2n - 1$. 2. **Understanding the prob

1. **Stating the problem:** We want to prove that there exist both rational and irrational numbers. 2. **Definition of rational numbers:** A rational number is any number that can

1. نبدأ بقراءة المسألة: لدينا عدد طبيعي $N$ مكون من رقمين. 2. المعطيات:

1. **Problem Statement:** We are given the recursive definition of Euclidean numbers: $$e_1 = 2$$

1. **Problem:** Solve the linear Diophantine equation $$172x + 20y = 1000$$ completely. 2. **Step 1: Find the gcd of 172 and 20.**

1. **State the problem:** Solve the linear Diophantine equation $$172x + 20y = 1000$$ for integers $x$ and $y$. 2. **Recall the condition for solutions:** A linear Diophantine equa

1. The problem is to solve for the value of number 19, which is ambiguous as stated. Assuming you want to understand the number 19 or solve a problem involving 19, please clarify.

1. **Problem Statement:** Investigate the last digits of integers raised to powers, specifically focusing on the pattern of the last one or two digits. 2. **Understanding the Probl

1. 問題陳述： 已知A同B都係21600嘅正因數，且A>B。求有幾多種選擇A同B嘅方法，使得兩個數嘅最大公因數（GCD）係1。

1. **Problem:** Determine if the numbers 51, 87, and 59 are prime or not prime. 2. **Recall:** A prime number is a number greater than 1 that has no positive divisors other than 1

1. The user asks: "what is the name of the proof for 3". 2. This question is ambiguous because "proof for 3" is not a standard mathematical phrase.

1. **Problem statement:** For each $n = 84$ and $n = 88$, find the smallest integer multiple of $n$ whose base 10 representation consists entirely of digits 6 and 7. 2. **Approach: