Number Theory

There are exactly 25 prime numbers between 1 and 100, starting from 2 and ending with 97. You can systematically count them or use the Sieve of Eratosthenes to identify all primes in this range.

There are only two ways to express 18 as the sum of two primes: 5 + 13 and 7 + 11. This is a specific case of Goldbach's conjecture, which states that every even number greater than 2 can be expressed as the sum of two primes.

The number 2 is the only even prime number. All other even numbers are divisible by 2 and therefore cannot be prime. Thus, there is exactly one even prime number in any range.

The series consists of consecutive prime numbers: 41, 43, 47. The next prime number after 47 is 53. 49 is divisible by 7, 57 is divisible by 3, and 59 is the prime after 53.

The prime numbers up to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. Counting them results in exactly 15 prime numbers.

We need to find pairs of prime numbers $(p, q)$ such that $p + q = 18$. The pairs are $(5, 13)$ and $(7, 11)$. Since there are exactly two such pairs, option 20824 is correct.

The smallest two-digit prime is 11. The greatest two-digit square is 81 (9^2). Their sum is 11 + 81 = 92.

Numbers that are neither prime nor even are odd composite numbers. The first is 1 (by definition here), the next odd numbers are 3 (prime), 5 (prime), 7 (prime), and 9 (composite). Thus, 9 is the second.

The Fibonacci sequence continues: 21, 34, 55, 89, 144... Of these, 21, 34, 55 are not prime. The number 89 is prime and is the next prime Fibonacci number after 13. 79 is not a Fibonacci number.

Let the four primes in ascending order be p, q, r, s. From the products: p × q × r = 385 = 5 × 7 × 11, so p=5, q=7, r=11. For the last three: q × r × s = 1001 = 7 × 11 × 13. This confirms the pattern and gives s = 13 as the largest prime.

When an odd number is added to itself an odd number of times, we're essentially calculating odd × odd. The product of two odd numbers is always odd. For example, if the odd number is 3 and we add it 3 times: 3 + 3 + 3 = 9, which is odd. This pattern holds for all odd numbers.

If the sum of two primes x and y is odd, one must be 2 (the only even prime) and the other must be odd. So we have 2 + odd = odd. The product is 2 × odd, which is always divisible by 2. For example: if x=2 and y=5, sum=7 (odd), product=10, which is divisible by 2.

Set H contains three-digit numbers where all digits (hundreds, tens, units) are prime digits (2, 3, 5, 7). There are 4 × 4 × 4 = 64 such numbers. When we add 5 to each, we must check which results still have all prime digits. Through systematic verification checking the units digit transformation (only endings of 2→7 and 7→2 work, while 3→8 and 5→0 fail) and accounting for carries affecting tens and hundreds places, exactly 20 numbers maintain all prime digits after adding 5.