🔢 number theory

1. **State the problem:** We are given a prime number $p$ such that $$p^2 \equiv 1.$$ We need to find which value of $p$ from the options satisfies this condition. 2. **Understand

1. The problem is to explain the number 7. 2. Seven is a natural number that comes after 6 and before 8.

1. **Stating the problem:** We want to find subsets of the digits of the number 141 that sum to the numbers 17, 32, 48, 60, and 64. 2. **Understanding subsets and sums:** A subset

1. **Stating the problem:** We need to use the numbers 1 through 9 to form a sum of 131, and from this sum, identify how to get the numbers 4, 16, 41, 48, and 66. 2. **Understandin

1. **Stating the problem:** We want to understand how to get the numbers 4, 16, 41, 48, and 66 from the number 131 using numbers from 01 through 69. 2. **Analyzing the problem:** T

1. **State the problem:** Prove that for any non-negative integer $n$, the expression $3^{2n} + 7$ is divisible by 8. 2. **Formula and approach:** We want to show that $3^{2n} + 7

1. **State the problem:** Prove that for any non-negative integer $n$, the expression $32n + 7$ is divisible by 8. 2. **Recall divisibility rules and formulas:** A number $a$ is di

1. **State the problem:** Prove that for any non-negative integer $n$, the expression $32n + 7$ is divisible by 8. 2. **Recall divisibility rules and formulas:** A number $a$ is di

1. The problem is to understand the number 18 and its properties. 2. The number 18 is a positive integer.

1. The problem is to find the next possible combination number after the given list of numbers. 2. Since the numbers appear to be arbitrary and not following a simple arithmetic or

1. The problem is unclear as stated, but it seems you want to use the digits 1, 1, 2, and 2 each exactly once, possibly to form a number or expression with one decimal point. 2. Le

1. **State the problem:** We want to verify the integral solution $(X,Y,Z,W) = (1484801, 1203120, 1169407, 1157520)$ for the equation $$X^4 + 2Y^4 = Z^4 + 4W^4.$$ 2. **Recall the e

1. **State the problem:** Find the remainder when $7^{222}$ is divided by 100. 2. **Formula and rules:** We use Euler's theorem which states that if $a$ and $n$ are coprime, then

1. **State the problem:** Find all positive integers $x, y, z$ such that $$x^2 + y^2 = z^2$$

1. **State the problem:** Find the remainder when $7^{222}$ is divided by 100. 2. **Formula and theorem:** We use Euler's theorem which states that if $a$ and $n$ are coprime, then

1. **State the problem:** Find all positive integers $x, y, z$ such that $$x^2 + y^2 = z^2$$

1. **بيان المسألة:** نريد إثبات أنه يمكن تمثيل أي مبلغ أكبر من 7 باستخدام قطع نقود معدنية من فئتي 3 و5. 2. **قاعدة الاستقراء:** نثبت صحة العلاقة للأعداد 8, 9, 10, 11, 12 كحالات أسا

1. **State the problem:** We are given the statement $5 \mid 8$ and asked to analyze it. 2. **Understand the notation:** The symbol $a \mid b$ means "$a$ divides $b$", i.e., $b$ is

1. The problem asks for the 6th perfect number. 2. A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself).

1. The problem asks to find counterexamples to the statement: "All prime numbers are odd." 2. Recall the definition: A prime number is a natural number greater than 1 that has no p

1. **State the problem:** We need to find a three-digit number with three different digits, where the ones place digit is 6, and the sum of the digits is 0. 2. **Analyze the proble