🔢 number theory

1. **Problem Statement:** We have three types of chocolate bags containing 6, 9, and 20 chocolates respectively. We want to find the largest number of chocolates that cannot be obt

1. Problem: Determine the remainder when $2026^{2026}$ is divided by 17. 2. Formula and rules: We use modular arithmetic and Fermat's Little Theorem which states that for a prime $

1. Problem: Does 17 divide each of these numbers? a) 68 b) 84 c) 357 d) 1001 2. To check if 17 divides a number $n$, we verify if $n \mod 17 = 0$.

1. **Problem:** Does 17 divide each of these numbers? a) 68 b) 84 c) 357 d) 1001 2. **Formula and rule:** An integer $a$ divides another integer $b$ (written $a \mid b$) if there e

1. **State the problem:** Find the exponent of 3 in the prime factorization of 2025. 2. **Recall the prime factorization process:** To find the exponent of a prime in a number, rep

1. **Problem statement:** We have a positive integer sequence $(t_n)_{n\geq 1}$ defined by the recurrence relation $$t_{n+2} = \frac{t_n + t_{n+1}}{\gcd(t_n, t_{n+1})}$$

1. The Riemann Hypothesis is a famous unsolved problem in mathematics concerning the zeros of the Riemann zeta function $\zeta(s)$.\n\n2. The hypothesis states that all non-trivial

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:**