The numbers 13 and 31 are prime numbers. Both these numbers have only two factors, 1 and themselves. Prime numbers cannot be divided evenly by any other number. 4. Write down separately the prime and composite numbers less than 30. Prime numbers less than 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Composite numbers less than 30: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28. 5. Find such pairs of prime numbers up to 100. Pairs of prime numbers whose difference is 2: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73). 6. What is the greatest prime number between 1 and 10? The greatest prime number between 1 and 10 is 7. 7. Express the following as the sum of two odd primes. (a) 44 = 19 + 25 (b) 36 = 17 + 19 (c) 24 = 11 + 13 (d) 18 = 7 + 11 8. Give three pairs of prime numbers whose difference is 2. Three pairs of prime numbers whose difference is 2: (3, 5), (5, 7), (11, 13). [Remark: Two prime numbers whose difference is 2 are called twin primes.] 9. Which of the following numbers are prime? (a) 23 is a prime number. (b) 51 is not a prime number. (c) 37 is a prime number. (d) 26 is not a prime number. 10. Write seven consecutive composite numbers less than 100 so that there is no prime number between them. 90, 91, 92, 93, 94, 95, 96.
The Correct Answer and Explanation is:
Here are the corrected answers with explanations:
7. Express the following as the sum of two odd primes:
(a) 44 = 17 + 27 ❌ (Incorrect, 27 is not prime)
Corrected: 44 = 23 + 21 ❌ (21 is not prime)
Corrected: 44 = 13 + 31 ✅
(b) 36 = 17 + 19 ✅
(c) 24 = 11 + 13 ✅
(d) 18 = 7 + 11 ✅
So the correct answers are:
(a) 44 = 13 + 31
(b) 36 = 17 + 19
(c) 24 = 11 + 13
(d) 18 = 7 + 11
Explanation
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. For example, 2, 3, 5, and 7 are all prime numbers. A composite number is a natural number greater than 1 that is not prime, meaning it has more than two distinct positive divisors. For example, 4, 6, 8, and 9 are composite numbers because they have factors other than 1 and themselves.
In the given questions, we explore several properties of prime numbers, including identifying them, recognizing twin primes, and expressing even numbers as the sum of two odd primes.
For instance, one question asked to express certain even numbers as the sum of two odd primes. This is related to the Goldbach Conjecture, which states that every even number greater than 2 can be expressed as the sum of two primes. While it remains unproven for all numbers, it holds true for all even numbers up to very high limits. For example, 44 can be expressed as 13 plus 31, both of which are prime numbers.
Another important concept is twin primes. Twin primes are pairs of prime numbers that have a difference of 2. Examples include (3, 5), (5, 7), and (11, 13). These primes are interesting to mathematicians because they are closely spaced, and it is still an open question whether there are infinitely many twin prime pairs.
Understanding the classification and behavior of primes and composites is fundamental in number theory and helps in developing strong mathematical reasoning skills.
