Q.1. What are prime numbers? List all primes between 1 and 30.
Ans: Those numbers with only two factors, i.e., 1 and the number itself, are known as prime numbers.
Examples: 2, 3, 5, 7, 11 and 13
The prime numbers between 1 and 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
Q.2. Write all prime numbers between:
(i) 10 and 50
(ii) 70 and 90
(iii) 40 and 85
(iv) 60 and 100
Ans:
(i) The prime numbers between 10 and 50 are 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.
(ii) The prime numbers between 70 and 90 are 71, 73, 79, 83 and 89.
(iii) The prime numbers between 40 and 85 are 41, 43, 47, 53, 59, 61, 67, 71, 73, 79 and 83.
(iv) The prime numbers between 60 and 100 are 61, 67, 71, 73, 79, 83, 89 and 97.
Q.3. What is the smallest prime number? Is it an even number?
Ans: The number 2 is the smallest prime number.
It is an even prime number. Except 2, all the other even numbers are composite numbers.
Q.4. What is the smallest odd prime? Is every odd number a prime number? If not, give an example of an odd number which is not prime.
Ans: The smallest odd prime number is 3.
No, every odd number is not a prime number. For example, 9 is an odd number but it is not a prime number because its three factors are 1, 3 and 9.
Q.5. What are composite numbers? Can a composite number be odd? If yes, write the smallest odd composite number.
Ans: A number which has more than two factors is called a composite number.
For example, the numbers 4, 6, 8, 9, 10 and 15 are composite numbers.
Yes, a composite number can be an odd number. The smallest odd composite number is 9.
Q.6. What are the twinprimes? Write all pairs of twinprimes between 50 and 100.
Ans: Twin primes: Two prime numbers are said to be twin primes if there is only one composite number between them.
For example, (3, 5) and (5, 7) are twin primes.
Twin primes between 50 and 100 are (59, 61) and (71, 73).
Q.7. What are coprimes? Give examples of five of coprimes. Are coprimes always prime? If no, illustrate your answer by an examples.
Ans: Two numbers are said to be coprimes if they do not have any common factor other than 1.
For example, (2, 3), (3, 4), (4, 5), (5, 7), and (13, 17) are coprimes.
Two coprime numbers need not be both prime numbers.
e.g., (3, 4), (6, 7) and (4, 13).
Q.8. Which of the following pairs are always coprimes?
(i) two prime numbers
(ii) one prime and one composite number
(iii) two composite numbers
Ans:
(i) Two prime numbers are always coprimes to each other.
Example: 7 and 11 are coprimes to each other.
(ii) One prime and one composite number are not always coprime.
Example: 3 and 21 are not coprimes to each other.
(iii) Two composite numbers are not always coprimes to each other.
Example: 4 and 6 are are not coprimes to each other.
Q.9. Express each of the following as a sum of two or more primes?
(i) 13
(ii) 130
(iii) 180
Ans: We can write the given numbers as the sums of two or more primes as follows:
(i) 13 = 11 + 2
(ii) 130 = 59 + 71
(iii) 180 = 139 + 17 + 11 + 13 or 79 + 101
Q.10. Express each of the following numbers as the sum of two odd primes:
(i) 36
(ii) 42
(iii) 84
Ans: We can express the given numbers as the sums of two odd primes as follows:
(i) 36 = 7 + 29 or 17 + 19
(ii) 42 = 5 + 37 or 13 + 29
(iii) 84 = 17 + 67 or 23 + 61
Q.11. Express each of the following numbers as the sum of three odd prime numbers:
(i) 31
(ii) 35
(iii) 49
Ans: We can express the given numbers as the sums of three odd prime numbers as follows:
(i) 31 = 5 + 7 + 19 or 31 = 11 + 13 + 7
(ii) 35 = 5 + 7 + 23 or 35 = 17 + 13 + 5
(iii) 49 = 3 + 5 + 41 or 49 = 7 + 11 + 31
Q.12. Express each of the following numbers as the sum of twin primes:
(i) 36
(ii) 84
(iii) 120
Ans: We can express the given numbers as the sums of twin primes which are as follows:
(i) 36 = 17 + 19
(ii) 84 = 41 + 43
(iii) 120 = 59 + 61
Q.13. Find the possible missing twins for the following numbers so that they become twin primes:
(i) 29
(ii) 89
(iii) 101
Ans:
(i) The possible missing twins for 29 are 27 and 31. Since 31 is a prime and 27 is not, 31 is the missing twin.
(ii) The possible missing twins for for 89 are 87 and 91. Since 87 and 91 are not primes, 89 has no twin.
(iii) The possible missing twins for 101 are 99 and 103. Since 103 is a prime and 99 is not, 103 is the missing twin.
Q.14. A list consists of the following pairs of numbers:
(i) 51, 53; 55, 57; 59, 61; 63, 65; 67, 69; 71, 73
Categorize them as pairs of:
(i) coprimes
(ii) primes
(iii) composites
Ans: (i) Coprimes: Two natural numbers are said to be coprime numbers if they have 1 as their only common factor.
Hence, all the given pairs of numbers are coprimes.
(ii) Primes: Natural numbers which have exactly two distinct factors, i.e., 1 and the number itself are called prime numbers.
Hence, (59, 61) and (71, 73) are pairs of prime numbers.
(iii) Composite numbers: Natural numbers which have more than two factors are called composite numbers.
Hence, (55, 57) and (63, 65) are pairs of composite numbers.
Q.15. For a number, greater than 10, to be prime what may be the possible digit in the unit's place?
Ans: For a number (greater than 10) to be a prime number, the possible digit in the unit's place may be 1, 3, 7, or 9.
Example: 11, 13, 17, and 19 are prime numbers greater than 10.
Q.16. Write seven consecutive composite numbers less than 100 so that there is no prime number between them.
Ans: The required seven consecutive composite numbers are 90, 91, 92, 93, 94, 95 and 96.
Q.17. State true (T) of false (F):
(i) The sum of primes cannot be a prime.
(ii) The product of primes cannot be a prime.
(iii) An even number is composite.
(iv) Two consecutive numbers cannot be both primes.
(v) Odd numbers cannot be composite.
(vi) Odd numbers cannot be written as sum of primes.
(vii) A number and its successor are always coprimes.
Ans:
(i) False.
2 + 3 = 5 which is a prime number.
(ii) True.
The product of prime numbers is always a composite number.
(iii) False.
The even number 2 is not a composite number.
(iv) False.
2 and 3 are consecutive numbers and are also prime numbers.
(v) False.
9 is an odd number but it is a composite number as its factors are 1, 3 and 9.
(vi) False.
9 is an odd number: 9 = 7 + 2 where 7 and 2 are prime numbers.
(vii) True.
A number and its successor have only one common factor (i.e., 1).
Q.18. Fill in the blanks in the following:
(i) A number having only two factors is called a .......................... .
(ii) A number having more than two factors is called a ..................... .
(iii) 1 is neither .................. nor ........................... .
(iv) The smallest prime number is .................................. .
(v) The smallest composite number is .................................. .
Ans:
(i) A number having only two factors is called a prime number.
(ii) A number having more than two factors is called a composite number.
(iii) 1 is neither composite nor prime.
(iv) The smallest prime number is 2.
(v) The smallest composite number is 4.
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