🔢 number theory

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. *

1. **State the problem:** We want to determine if exact change can be made for the amounts 29, 19, 23, and 33 using an infinite supply of coins with denominations 6, 10, and 15. 2.

1. **State the problem:** Find the highest power of 20 that divides $50!$. 2. **Prime factorization of 20:**

1. **Problem statement:** We start with numbers 1, 2, 3, ..., 100 on the board. At each step, two numbers $a$ and $b$ are chosen and replaced by $|a - b|$. This continues until onl

1. **State the problem:** We want to determine which of the amounts 19, 29, 23, and 33 can be formed exactly using any number of coins of denominations 6, 10, and 15. 2. **Understa

1. **State the problem:** Find the highest power of 20 that divides 50!. 2. **Prime factorize 20:**

1. **State the problem:** We want to determine if exact change can be made for the amounts 23, 29, 19, and 33 using an infinite supply of coins with denominations 6, 10, and 15. 2.

1. **State the problem:** We want to prove by mathematical induction that the number $$U_n = 2^{6n} + 3^{2n-2}$$ is divisible by 5 for all positive integers $n$. 2. **Base case ($n

1. The problem asks if -9999 can be considered the smallest 4-digit number. 2. A 4-digit number is any integer from 1000 to 9999 or from -9999 to -1000 if considering negative numb