Number Theory and Algebra

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. 17 satisfies this condition, while the other options do not.

Using the Euclidean algorithm, we have: 36 = 24 * 1 + 12, 24 = 12 * 2 + 0. Therefore, the GCD of 24 and 36 is 12.

To solve the modular equation, we can multiply both sides by the modular inverse of 3 modulo 7. Since 3 * 5 ≡ 1 (mod 7), the modular inverse of 3 modulo 7 is 5. Multiplying both sides by 5, we get x ≡ 2 (mod 7). Therefore, the solution is x = 2.

We can factorize the polynomial using synthetic division or by grouping terms. By grouping, we have: (x^3 - 2x^2) - (5x - 6) = x^2(x - 2) - 1(x - 2) = (x^2 - 1)(x - 2) = (x - 1)(x + 1)(x - 2).

The order of an element in a group is the smallest positive integer n such that a^n = e, where a is the element and e is the identity element. In the group (Z_7, +), the identity element is 0. We have: 3^1 = 3, 3^2 = 6, 3^3 = 2, 3^4 = 5, 3^5 = 1. Therefore, the order of the element 3 is 3.

A field is a set with two binary operations, addition and multiplication, that satisfy certain properties. The set of complex numbers (C) with the usual operations of addition and multiplication forms a field, while the other options do not.

The equation x^2 + 2x + 1 = 0 is a quadratic equation. Using the quadratic formula, we have: x = (-2 ± √(2^2 - 4 * 1 * 1)) / (2 * 1) = -1 ± √3i. Therefore, there are two solutions to the equation in the field of complex numbers.

A cyclic group is a group that is generated by a single element. The set of integers (Z) with the usual operation of addition forms a cyclic group, as it is generated by the element 1. The other options are not cyclic groups.

The multiplicative inverse of an element in a field is the element that, when multiplied by the given element, results in the identity element. In the field of integers modulo 11, the identity element is 1. We have: 7 * 8 ≡ 1 (mod 11). Therefore, the multiplicative inverse of 7 in the field of integers modulo 11 is 8.

A prime ideal in a ring is an ideal that is also a prime ideal. In the ring of integers, the ideal (7) is a prime ideal because it is a maximal ideal and it is also a prime ideal.

The group U(10) consists of all the integers between 1 and 10 that are relatively prime to 10. These integers are 1, 3, 7, and 9. Therefore, the number of elements in the group U(10) is 4.

A subgroup of a group is a non-empty subset of the group that is closed under the group operation. The set of integers {0, 6} with the usual operation of addition forms a subgroup of the group (Z_12, +), as it is non-empty, closed under addition, and contains the identity element 0.

The characteristic of a field is the smallest positive integer n such that 1 + 1 + ... + 1 (n times) = 0. In the field GF(2^8), the smallest positive integer n such that 1 + 1 + ... + 1 (n times) = 0 is 2. Therefore, the characteristic of the field GF(2^8) is 2.

A normal subgroup of a group is a subgroup that is invariant under conjugation by every element of the group. The set of even permutations in the group (S_3, *) forms a normal subgroup, as it is invariant under conjugation by every element of the group.

The number of generators of a cyclic group of order n is φ(n), where φ is the Euler totient function. For n = 15, φ(15) = 8. Therefore, the number of generators of the cyclic group of order 15 is 4.