🔢 number theory

1. **State the problem:** Calculate $254^{94} \bmod 160$ efficiently by hand. 2. **Simplify the base modulo 160:** Since $254 > 160$, reduce it first:

1. Асуудлыг тодорхойлно: 4-т хуваагдах бүх тоог сонгох. 2. 4-т хуваагдах тоо гэдэг нь 4-өөр хуваагдаж үлдэгдэл 0 гардаг тоо юм.

1. **Problem statement:** Given that $\gcd(a^n,b^n)=1$ and $\gcd(a,b^n)=1$, prove by induction that $\gcd(a^{n+1}+1,b^{n+1}+1)=1$. 2. **Base case (n=1):** We need to show $\gcd(a^{

1. **Problem statement:** Given that $\gcd(a^n,b^n) = 1$ and $\gcd(a,b^n) = 1$, show that $\gcd(a^{n+1}, b^{n+1}) = 1$. 2. **Recall the properties of gcd:**

1. لنفترض أن السؤال يتعلق بكيفية معرفة أن المتغير $d$ يقسم عددًا ما أو تعبيرًا رياضيًا. 2. في الرياضيات، نقول أن $d$ يقسم عددًا $n$ إذا كان هناك عدد صحيح $k$ بحيث أن $n = d \times

1. **State the problem:** We want to prove that if $\gcd(a,b) = 1$, then $\gcd(a+b, ab) = 1$. 2. **Recall the definition:** $\gcd(x,y)$ is the greatest positive integer that divide

1. **State the problem:** We want to prove that if $\gcd(a,b) = 1$, then $\gcd(a+b, ab) = 1$. 2. **Recall the definition:** $\gcd(x,y)$ is the greatest positive integer that divide

1. **Problem statement:** Given integers $a$, $b$, and $c$, none of which are zero simultaneously, and $d = \gcd(a,b,c)$, show that $$d = \gcd(\gcd(a,b), c) = \gcd(a, \gcd(b,c)) =

1. **State the problem:** We want to show that for integers $a, b, c$ (not all zero) and $d = \gcd(a, b, c)$, the following equalities hold: $$d = \gcd(\gcd(a, b), c) = \gcd(a, \gc

1. The problem is to find the greatest common divisor (GCD) of the numbers 312, 260, and 156. 2. First, find the prime factorization of each number:

1. **State the problem:** We need to prove that for any positive integer $n$, the expression $7^n - 1$ is divisible by 6. 2. **Rewrite the problem:** To say $7^n - 1$ is divisible

1. **Problem Statement:** Divide the hexadecimal number $\text{(13AF9)}_{16}$ by $\text{(9A)}_{16}$ and find the quotient and remainder. 2. **Convert divisor to decimal:**

1. **Problem statement:** We want to find which postage amounts can be made using only five-cent and six-cent stamps. Specifically, is there a number $N$ such that for every $n \ge

1. **State the problem:** We need to find Clara's number which satisfies three conditions: - It is prime.

1. A composite number is a positive integer greater than 1 that has more than two positive divisors. 2. This means a composite number can be divided evenly by numbers other than 1

1. समस्या: यदि $3^{n+1} - an - b4$ विभाज्य है, जहाँ 0 और $b$ सह-अभाज्य हैं, तो $o > b$ का मान ज्ञात करें। 2. सबसे पहले, अभाज्यता की शर्त को समझते हैं। यहाँ $3^{n+1} - an - b4$ को क

1. The problem asks to identify which number among the options is a prime number. 2. Recall that a prime number is a natural number greater than 1 that has no positive divisors oth

1. Let's analyze each statement about the largest prime number. 2. Statement 1: "The largest prime number can be found using quantum computers." This is false because there is no l

1. The problem involves understanding the behavior of the function $A(n)$, which counts the number of prime numbers from 1 up to $n$. 2. We analyze each statement:

1. Euler's theorem states that if $n$ and $a$ are coprime (i.e., their greatest common divisor is 1), then: $$a^{\varphi(n)} \equiv 1 \pmod{n}$$

1. **State the problem:** We want to determine which amounts among 19, 33, 29, and 23 can be formed exactly using an infinite supply of coins with denominations 6, 10, and 15. 2. *