Number Theory and Diophantine Equations

9 is a composite number because it has factors other than 1 and itself (e.g., 3).

The GCD of 18 and 24 can be found using the Euclidean algorithm: 24 = 18 * 1 + 6; 18 = 6 * 3 + 0. Therefore, the GCD is 6.

Using modular arithmetic, we can calculate 7^25 mod 13 by raising 7 to the power of 25 mod 13: 7^25 mod 13 = (7^4)^6 * 7 mod 13 = 1 * 7 mod 13 = 3.

To solve the equation, we can express y in terms of x: y = (11 - 3x) / 5. Substituting different values of x, we find that x = 1 and y = 2 satisfy the equation.

A Pythagorean triple satisfies the equation a^2 + b^2 = c^2. (3, 4, 5) satisfies this equation: 3^2 + 4^2 = 5^2.

We can check each option: 2^2 + 1 = 5 (prime), 3^2 + 1 = 10 (not prime), 5^2 + 1 = 26 (not prime), 7^2 + 1 = 50 (not prime). Therefore, n = 2.

To be relatively prime to 100, a number must not share any common factors with 100. The prime factorization of 100 is 2^2 * 5^2. Therefore, any number that does not contain these factors is relatively prime to 100. There are 40 such numbers less than 100.

A Fermat prime is a prime number of the form 2^(2^n) + 1. 17 is a Fermat prime because 2^(2^2) + 1 = 17.

To be divisible by 100, n! must contain at least two factors of 5. The smallest positive integer that satisfies this condition is 25, as 25! contains two factors of 5.

The multiples of 7 less than 100 are: 7, 14, 21, ..., 98. The sum of this arithmetic sequence can be calculated using the formula: sum = (n/2) * (a1 + an), where n is the number of terms, a1 is the first term, and an is the last term. Plugging in the values, we get: sum = (14/2) * (7 + 98) = 840.

We can try different values of x and y to find solutions. Substituting x = 1, we get y^2 = 16, which gives y = 4. Therefore, one solution is (1, 4).

To be divisible by 3 but not by 5, a number must be divisible by 3 but not divisible by 15. The multiples of 3 less than 1000 are: 3, 6, 9, ..., 999. The multiples of 15 less than 1000 are: 15, 30, 45, ..., 990. Subtracting the multiples of 15 from the multiples of 3, we get the numbers that are divisible by 3 but not by 5. There are 266 such numbers.

The GCD of 270 and 195 can be found using the Euclidean algorithm: 270 = 195 * 1 + 75; 195 = 75 * 2 + 45; 75 = 45 * 1 + 30; 45 = 30 * 1 + 15; 30 = 15 * 2 + 0. Therefore, the GCD is 15.

Using modular arithmetic, we can calculate 11^100 mod 13 by raising 11 to the power of 100 mod 13: 11^100 mod 13 = (11^4)^25 * 11 mod 13 = 1 * 11 mod 13 = 1.

A Mersenne prime is a prime number of the form M_p = 2^p - 1, where p is a prime number. 31 is a Mersenne prime because 2^31 - 1 = 31.