🔢 number theory

1. **Stating the problem:** We have $p=3$ and the set

1. **Problem Statement:** We have $p=3$ and the set

1. We need to find the units digit of $4^{7022}$. 2. The units digit of powers of 4 cycle every 2: $4^1=4$, $4^2=16$ (units digit 6), $4^3=64$ (units digit 4), $4^4=256$ (units dig

1. **Stating the problem:** We are given two numbers, 661 and 4779, and their remainders when divided by 7.

1. **Problem statement:** Find the number of ordered triples $(a,b,c)$ of positive integers with $1 \leq a,b,c \leq 50$ such that $$\frac{\mathrm{lcm}(a,c) + \mathrm{lcm}(b,c)}{a+b

1. **Problem Statement:** We want to find how many possible values there are for the sum $a + b + c$ where $a$, $b$, and $c$ are positive integers such that their product $abc = 72

1. **Problem:** Show that $\gcd(3a + 5, 7a + 12) = 1$ for $a > 0$ and $a \in \mathbb{Z}$.\n\n2. **Formula and rules:** The greatest common divisor (gcd) of two integers can be foun

1. **Problem statement:** We need to find a whole number $n$ such that the product $n \times (n + 42)$ is a prime number. 2. **Recall the definition of a prime number:** A prime nu

1. The problem is to understand and apply the Euclidean algorithm to find the greatest common divisor (GCD) of two integers. 2. The Euclidean algorithm is based on the principle th

1. The Collatz Conjecture states: Start with any positive integer $n$. If $n$ is even, divide it by 2; if odd, multiply by 3 and add 1. Repeat the process. 2. The question is wheth

1. **Problem statement:** Given integers $c,d$ and the sequence $a_n = c^n + d$, prove that for any large integer $k$, there exists a positive integer $n$ such that in the list $a_

1. **Problem statement:** Given integers $c,d$ and the sequence $a_n = c^n + d$, prove that for any large integer $k$, there exists a positive integer $n$ such that in the list $a_

1. **Problem Statement:** We want to find how many possible values there are for the sum $a + b + c$ where $a$, $b$, and $c$ are positive integers such that $abc = 72$. 2. **Unders

1. **Problem statement:** Determine the pairs of integers $(x,y)$ such that the base 5 number $(xy)_5$ equals the base 9 number $(yx)_9$. 2. **Understanding the problem:**

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