🔢 number theory

1. **Problem:** Given the 11-digit number $21A3609727B$ is divisible by 44, find the sum of all possible values of $A + B$. 2. **Recall:** A number is divisible by 44 if and only i

1. **Problem:** Make the number 19 using only the numbers 7, 9, 13, and 20 with addition or subtraction. 2. **Approach:** We want to find integers $a, b, c, d$ such that:

1. **Problem statement:** We want to find a way to make the number 12 using the numbers 7, 9, 13, and 20. 2. **Approach:** We can try to express 12 as a combination of these number

1. مسئله: همه اعداد دو رقمی مضرب 3 را پشت سر هم می‌چینیم و عدد حاصل را بر 11 تقسیم می‌کنیم. باقی‌مانده تقسیم این عدد بر 11 را پیدا کنید. 2. اعداد دو رقمی مضرب 3 از 12 شروع شده و تا

1. مسئله: همه اعداد دو رقمی مضرب 3 را پشت سر هم می‌چینیم و عدد حاصل را بر 11 تقسیم می‌کنیم. باقی‌مانده تقسیم این عدد بر 11 را پیدا کنید. 2. اعداد دو رقمی مضرب 3 از 12 شروع شده و تا

1. **Problem Statement:** We have a long division where a six-digit number $X$ is divided by a three-digit number $Y$ to get a three-digit quotient $Z$ starting with 8.

1. **State the problem:** We need to find the base $x$ such that the number $110_x$ in base $x$ equals $40_5$ in base 5. 2. **Convert $40_5$ to decimal:** In base 5, $40_5 = 4 \tim

1. **Problem statement:** We want to prove that for all integers $n \geq 24$, there exist nonnegative integers $k$ and $l$ such that $$n = 5k + 7l.$$ 2. **Method:** We will use com

1. **问题陈述**：求解同余方程组： $$x \equiv 11 \pmod{11},\quad x \equiv 2 \pmod{12},\quad x \equiv 4 \pmod{13}$$

1. **State the problem:** Solve the congruence equation $$x \equiv 1694 \pmod{1716}$$. 2. **Understand the problem:** The congruence means that $x$ and $1694$ leave the same remain

1. **State the problem:** Solve the system of congruences using the Chinese Remainder Theorem (CRT): $$x \equiv 11 \pmod{11}$$

1. **Problem statement:** Solve the system of congruences using the Chinese Remainder Theorem (CRT): $$x \equiv 2 \pmod{11}$$

1. **Problem statement:** Solve the system of congruences using the Chinese Remainder Theorem (CRT): $$x \equiv 7 \pmod{15}$$

1. **Problem Statement:** We need to show that if we choose 101 numbers from the list 1 to 200, there must be at least two numbers among the chosen that have a highest common facto

1. The problem is unclear as "jodi digit" is not a standard math term. Assuming you want to understand what a "jodi digit" means in number theory or a related context. 2. In some c

1. The problem is to understand what the greatest common divisor (GCD) is. 2. The GCD of two or more integers is the largest positive integer that divides each of the integers with

1. **State the problem:** We need to find which number among 45, 60, 91, 105, and 330 is the product of exactly three distinct prime numbers. 2. **Recall the concept:** A product o

1. The problem is to determine which number among 51, 102, 153, 867, or 2601 fits a certain pattern or rule. 2. Since no explicit pattern is given, let's analyze the numbers for co

1. The problem is to convert the number \texttt{31five} from base 5 to base 10. 2. The formula to convert a number from base $b$ to base 10 is:

1. The problem asks us to identify which number among 6, 28, 10, and 12 is not a perfect number. 2. A perfect number is a positive integer that is equal to the sum of its proper di

1. The problem is to find all prime numbers between 1 and 100. 2. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.