Certainly! Let's address each part of the question step-by-step.
### a) How many prime numbers are there between 1 and 100?
First, we'll find all the prime numbers between 1 and 100. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Using the Sieve of Eratosthenes, we can list all the prime numbers in this range and count them.
The prime numbers between 1 and 100 are:
[tex]\[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 \][/tex]
Counting these, we find there are 25 prime numbers between 1 and 100.
### b) List all the prime numbers ending with 3 and 7.
Next, we need to identify prime numbers in the list above that end with the digits 3 and 7.
Prime numbers ending with 3:
[tex]\[ 3, 13, 23, 37, 43, 53, 73, 83 \][/tex]
Prime numbers ending with 7:
[tex]\[ 7, 17, 37, 47, 67, 97 \][/tex]
Combining both lists, the unique prime numbers ending with 3 or 7 are:
[tex]\[ 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97 \][/tex]
### List down all the sets of twin primes from 1 to 100.
Twin primes are pairs of prime numbers that have a difference of 2. In other words, if [tex]\(p\)[/tex] and [tex]\(p + 2\)[/tex] are both prime numbers, they are considered a twin prime pair.
From the list of prime numbers between 1 and 100, the twin primes are:
[tex]\[ (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73) \][/tex]
### Summary
- The number of prime numbers between 1 and 100 is 25.
- The prime numbers ending with 3 or 7 are: [tex]\[3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97\][/tex].
- The sets of twin primes from 1 to 100 are: [tex]\[(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73)\][/tex].
Fill in the blanks as per the result:
- f) 30: There seems to be no direct match for this within our steps, please recheck the given context.
- g) 78: There seems to be no direct match for this within our steps, please recheck the given context.
