🔢 number theory

1. مسئله: تعداد اعداد اول کوچکتر از 100 را پیدا کنید. 2. تعریف عدد اول: عدد اول عددی طبیعی بزرگتر از 1 است که فقط بر 1 و خودش بخش‌پذیر باشد.

1. **Problem Statement:** Find the digital root of a number by repeatedly summing its digits until a single digit (1 to 9) remains. 2. **Definition:** The digital root of a number

1. **Problem statement:** Find the set of solutions for the linear congruences: i. $x \equiv 3 \pmod{5}$

1. **State the problem:** Show that $11^9 + 9^{11}$ is divisible by 10. 2. **Recall the divisibility rule:** A number is divisible by 10 if its last digit is 0, which means the num

1. **Problem statement:** A number when divided by 30 leaves a remainder of 8. We need to find what number should be added to this number to make it a multiple of 6. 2. **Understan

1. **Stating the problem:** We need to identify which numbers among 153, 257, 408, and 441 are definitely not perfect squares without performing detailed calculations. 2. **Importa

1. **Problem Statement:** Find the possible one's digits of the square roots of the given numbers: 9801, 059856, 998001, 657666025. 2. **Important Rule:** The one's digit of a perf

1. **Problem Statement:** Find the digit $D$ such that the last digit of $(54D)^{100}$ is 1. 2. **Understanding the problem:** The last digit of a number raised to a power depends

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): $$x \equiv 7 \pmod{15}$$