Number Theory

The GCD of 18 and 24 is the largest positive integer that divides both numbers without leaving a remainder. By finding the prime factorization of each number, we have 18 = 2 * 3^2 and 24 = 2^3 * 3. The GCD is the product of the common prime factors raised to their lowest powers, which is 2 * 3 = 6.

A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. Among the given options, only 17 satisfies this condition. 12 is divisible by 2 and 3, 21 is divisible by 3 and 7, and 29 is prime.

Prime factorization involves expressing a number as a product of prime numbers. For 100, we can write it as 100 = 2 * 2 * 5 * 5 = 2^2 * 5^2.

Modular arithmetic involves calculations with integers modulo a fixed integer. In this case, we are working modulo 5. To find 7^3 mod 5, we can calculate 7^3 = 343, and then divide it by 5 to get the remainder. 343 ÷ 5 = 68 remainder 3. Therefore, 7^3 mod 5 = 3.

To find the smallest positive integer divisible by 2, 3, and 5, we need to find their least common multiple (LCM). The LCM of 2, 3, and 5 is 30, which is the smallest number that is divisible by all three numbers.

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Adding these numbers together, we get 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 71.

Goldbach's conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. It is one of the most famous unsolved problems in mathematics.

Euler's totient function φ(n) counts the number of positive integers less than or equal to n that are relatively prime to n. For n = 12, the numbers relatively prime to 12 are 1, 5, 7, and 11. Therefore, φ(12) = 4.

To find the smallest positive integer n such that n! is divisible by 100, we need to find the smallest n for which n! contains at least two factors of 5 and two factors of 2. The smallest number that satisfies this condition is 25, as 25! contains 6 factors of 5 and 12 factors of 2.

A perfect number is a positive integer that is equal to the sum of its proper divisors (divisors excluding the number itself). Among the given options, only 8128 is a perfect number, as its proper divisors are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, and 4096, and their sum is 8128.

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, typically starting with F(0) = 0 and F(1) = 1. To find F(10), we can use the recursive formula: F(n) = F(n-1) + F(n-2). Starting from F(0) = 0 and F(1) = 1, we can calculate F(10) = 55.

To find the greatest integer n such that 2^n divides 1000!, we need to determine the highest power of 2 that divides 1000!. We can do this by repeatedly dividing 1000! by 2 until the result is no longer divisible by 2. After performing this process, we find that 2^26 divides 1000!, but 2^27 does not. Therefore, the greatest integer n is 26.

A Mersenne prime is a prime number of the form M_p = 2^p - 1, where p is a prime number. Among the given options, only M7 = 2^7 - 1 = 127 is a prime number. Therefore, M7 is a Mersenne prime.

To find the sum of the first 100 prime numbers, we can use a loop or a pre-calculated list of prime numbers. Summing the first 100 prime numbers, we get 2 + 3 + 5 + 7 + 11 + ... + 541 = 24133.

The sum of all positive divisors of a number n is denoted by σ(n). For σ(n) = 2n, we need to find the smallest positive integer n such that the sum of its divisors is twice the number itself. The divisors of 6 are 1, 2, 3, and 6, and their sum is 12, which is twice the value of 6. Therefore, the smallest positive integer n is 6.