Number Theory

Mersenne primes are prime numbers of the form $2^n - 1$. Here, $3 (2^2-1)$, $7 (2^3-1)$, $31 (2^5-1)$, and $127 (2^7-1)$ all fit this definition. Fermat primes follow a different formula ($2^{2^n}+1$).

The statement is false because 2 is both an even number (divisible by 2) and a prime number (has exactly two factors: 1 and 2). It is the only even prime number, which is a well-known exception.

A multiperfect number (or k-perfect number) is a number where the sum of all divisors is an integer multiple of the number itself. These are also known as pluperfect numbers. Pseudoperfect numbers refer to a different property (semiperfect).

599 is a prime number because it has no divisors other than 1 and itself. Checking divisibility by primes up to sqrt(599) ≈ 24.5 (primes: 2, 3, 5, 7, 11, 13, 17, 19, 23) shows none divide 599 evenly. 899 = 29×31, 299 = 13×23, and 799 = 17×47 are all composite.

Whole numbers are the set of natural numbers plus zero {0, 1, 2, 3...}. Natural numbers typically start at 1, and prime numbers have specific divisibility rules.

By definition, 1 is the only positive integer that is neither prime nor composite. 2, 13, and 17 are all prime numbers because they have exactly two distinct factors. 1 has only one factor.

A prime number is defined as a natural number greater than 1 with exactly two distinct positive divisors: 1 and itself. The number 1 has only one divisor, so it is not prime. Two is the smallest prime number.

97 is a prime number as it has no divisors other than 1 and 97. 33 is divisible by 3 and 11; 81 is divisible by 3 and 9; 93 is divisible by 3 and 31. Thus, 97 is the only prime in the list.

A prime number has only two factors: 1 and itself. 31, 61, and 71 are prime. However, 91 is composite because $7 \times 13 = 91$. Therefore, 91 is the correct choice as it is not a prime number.

The smallest 3-digit number is 100. Checking numbers upwards: 101 is prime. Since 101 is not listed in the options (103, 107, 109), 'None' is the correct choice. 103, 107, and 109 are primes, but they are larger than 101.

Two is the only even prime number because all other even numbers are divisible by 2, making them composite. A prime number has exactly two factors: 1 and itself. Since any even number greater than 2 has at least three factors (1, 2, and itself), only 2 satisfies the prime number condition among even numbers.

The first three prime numbers are 2, 3, and 5. Their sum is 2 + 3 + 5 = 10. Remember that 1 is not considered a prime number in modern mathematics. This is a fundamental concept in number theory.

All even numbers are divisible by 2. For an even number to also be prime, it can only have 1 and itself as factors, which means 2 is the only even number that satisfies both conditions.

The mean of two consecutive primes is (p1 + p2)/2. For all primes greater than 2, primes are odd, so their sum is even and the mean is an integer. However, that integer always lies between the two primes; since there are no primes between consecutive primes, the mean cannot be prime.