🔢 number theory

1. **State the problem:** Solve the congruence equation $$x^2 \equiv -1 \pmod{17}$$ for integers $$x$$ such that $$1 \leq x \leq 17$$. 2. **Rewrite the congruence:** Since $$-1 \eq

1. **Problem statement:** We need to find digits $x$ and $y$ such that the number $42x4y$ is divisible by 72. 2. **Key fact:** A number is divisible by 72 if and only if it is divi

1. **Problem statement:** Find the greatest number which divides 70 and 125 leaving remainders 5 and 8 respectively. 2. **Understanding the problem:** If a number $d$ divides 70 le

1. The problem is to convert the number 80 into Roman numerals. 2. Roman numerals use letters to represent values: I=1, V=5, X=10, L=50, C=100, D=500, M=1000.

1. The problem is to convert the Roman numeral XIX into a number. 2. Roman numerals are based on combinations of letters from the Latin alphabet: I, V, X, L, C, D, and M.

1. The problem is to convert the number 109 into Roman numerals. 2. Roman numerals use letters to represent values: I=1, V=5, X=10, L=50, C=100, D=500, M=1000.

1. The problem is to convert the number 206 into Roman numerals. 2. Roman numerals use letters to represent values: I=1, V=5, X=10, L=50, C=100, D=500, M=1000.

1. The problem is to convert the given number into Roman numerals. 2. Roman numerals use letters to represent values: I=1, V=5, X=10, L=50, C=100, D=500, M=1000.

1. **Problem Statement:** We are given two co-prime numbers whose product is 553. We need to find their Highest Common Factor (HCF). 2. **Key Concept:** Two numbers are co-prime if

1. **State the problem:** Find the Highest Common Factor (HCF) of 112, 224, and 336. 2. **Formula and rules:** The HCF of numbers is the largest number that divides all of them wit

1. The problem asks if there are any whole number solutions $(x, y, z)$ to the equation $$x^n + y^n = z^n$$ where $n$ is a whole number greater than 2. 2. This is a famous problem

1. The problem is to create a mathematical formula that predicts the prime number sequence, starting from the 100th prime number. 2. Prime numbers are numbers greater than 1 that h

1. **Stating the problem:** We need to determine which of the given numbers is a multiple of 45. 2. **Formula and rules:** A number is a multiple of 45 if it is divisible by both 9

1. **Find the number of positive integers \( \leq 3000 \) divisible by 3, 5 or 7.** We use the Inclusion-Exclusion Principle.

1. **Stating the problem:** Classify the numbers 2.7, \(\frac{2}{4}\), and 2 \(\frac{8}{9}\) into the sets: Real Numbers, Irrational Numbers, Rational Numbers, Integers, Whole Numb

1. **State the problem:** We need to find the value of $d$ such that $5d \equiv 1 \pmod{96}$. This means $5d$ leaves a remainder of 1 when divided by 96. 2. **Formula and concept:*

1. The problem is to find the sequence using Euler's theorem. 2. Euler's theorem states that if $a$ and $n$ are coprime (i.e., $\gcd(a,n)=1$), then:

1. **Stating the problem:** We want to identify prime numbers easily. 2. **Definition:** A prime number is a natural number greater than 1 that has no positive divisors other than

1. **Problem Statement:** We need to find the multiplicative inverse of $a=5$ modulo a prime number $p=17$. This means finding an integer $x$ such that: $$5 \times x \equiv 1 \pmod

1. Problem: Explain why $-(m+1)!(p-m-2)!\equiv(-1)^{m+1}\pmod{p}$ becomes $(m+1)!(p-m-2)!\equiv(-1)^{m+2}\pmod{p}$ and prove that for a prime $p$ and $0\le k\le p-1$ we have $k!(p-

1. **Find a positive integer $x$ such that** $$x \equiv 2 \pmod{4}, \quad 2x \equiv 3 \pmod{9}, \quad 7x \equiv 1 \pmod{11}.$$