🔢 number theory

1. **Problem Statement:** We need to show that if we choose 101 numbers from the list 1 to 200, there must be at least two numbers among the chosen that have a highest common facto

1. The problem is unclear as "jodi digit" is not a standard math term. Assuming you want to understand what a "jodi digit" means in number theory or a related context. 2. In some c

1. The problem is to understand what the greatest common divisor (GCD) is. 2. The GCD of two or more integers is the largest positive integer that divides each of the integers with

1. **State the problem:** We need to find which number among 45, 60, 91, 105, and 330 is the product of exactly three distinct prime numbers. 2. **Recall the concept:** A product o

1. The problem is to determine which number among 51, 102, 153, 867, or 2601 fits a certain pattern or rule. 2. Since no explicit pattern is given, let's analyze the numbers for co

1. The problem is to convert the number \texttt{31five} from base 5 to base 10. 2. The formula to convert a number from base $b$ to base 10 is:

1. The problem asks us to identify which number among 6, 28, 10, and 12 is not a perfect number. 2. A perfect number is a positive integer that is equal to the sum of its proper di

1. The problem is to find all prime numbers between 1 and 100. 2. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

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