Class 6 Playing with Numbers Exercise 2.3-1

\begin{array}{l} \text {Q1. What are prime numbers? List all primes between 1 and 30.} \\ \\\text {Sol. A number is called a prime number if it has no factor } \\ \text { other than 1 and the number itself. } \\ \text {Like 2,3,5,7,11 and 13 are prime numbers.} \\ \text {The list of primes between 1 and 30 are } \\ \text {2,3,5,7,11,13,17,19,23 and 29 } \\ \\\text {Q2. Write all prime numbers between:} \\ \text {(i) 10 and 50 } \\ \text {(ii) 70 and 90 } \\ \text {(iii) 40 and 85 } \\ \text {(iv) 60 and 100 } \\ \\\text {Sol (i) The prime numbers between 10 and 50 are } \\ \text {11,13,17,19,23,29,31,37,41,43 and 47 } \\ \text {(ii) The prime numbers between 70 and 90 are } \\ \text {71,73,79,83 and 89 } \\ \text {(iii) The prime numbers between 40 and 85 are } \\ \text {41,43,47,53,59,61,67,71,73,79 and 83 } \\ \text {(iv) The prime numbers between 60 and 100 are } \\ \text {61,67,71,73,79,83,89 and 97 } \\ \\\text {Q3. What is the smallest prime number? Is it an even number? } \\ \\\text {Sol. 2 is the smallest prime number.} \\ \text {And 2 is also an even prime number because it is divisible by 2.} \\ \\\text {Q4. What is the smallest odd prime? Is every odd number a prime number? } \\ \text { If not, give an example of an odd number which is not prime. } \\ \\\text {Sol. 3 is the smallest odd prime number. } \\ \text {No, every odd number is not a prime number. } \\ \text {For example, 9 is an odd number having factors 1, 3 and 9 } \\ \text {and is not a prime number.} \\ \\\text {Q5. What are composite numbers? Can a composite number be odd? } \\ \text {If yes, write the smallest odd composite number.} \\ \\\text {Sol. A number is called a composite number if it has at least } \\ \text {one factor other than 1 and the number itself.} \\ \text {For example – 4,6,8,9,10 and 15 are composite numbers.} \\ \text {Yes, a composite number can be odd and 9 is the smallest odd composite number.} \\\text {Q6. What are twin-primes? Write all pairs of twin-primes between 50 and 100.} \\ \\\text {Sol. Two prime numbers are known as twin-primes if there } \\ \text {is only one composite number between them. } \\ \text {For example – (3, 5) , (5,7) , (11,13) , (17,19) , (29,31) , (41,43) , (59,61) and (71,73) } \\ \text {are the pairs of twin-primes between 1 and 100} \\ \\\text {Q7. What are co-primes? Give examples of five pairs of co-primes. } \\ \text {Are co-primes always prime? If no, illustrate your answer by an example. } \\ \\\text {Sol. Two numbers are said to be co-prime if they do not have a common factor other than 1.} \\ \text {Five pair of co-prime numbers are } \\ \text { (2,3) ; (3,4) ; (4,5) ; (5,6) ; (6,7) } \\ \text {No, It not necessary that co-primes are always prime.} \\ \text { (15,16) are co-prime but both of the numbers are not prime.} \\ \\\text {Q8. Which of the following pairs are always co-primes?} \\ \text {(i) two prime numbers } \\ \text {(ii) one prime and one composite number } \\ \text {(iii) two composite numbers } \\ \\\text {Sol. (i) Two prime numbers are always co-prime to each other.} \\ \text {The numbers 7 and 11 are co-prime to each other.} \\ \\\text {(ii) One prime and one composite number are not always co-prime. } \\ \text {For example – The numbers 3 and 21 are not co-prime to cach other.} \\ \\\text {(iii) Two composite numbers are not always co-prime to each other.} \\ \text {For example – 4 and 6 are not co-prime to each other.} \\ \\\text {Q9. Express each of the following as a sum of two or more primes: } \\ (i) \quad 13 \\ (ii) \quad 130 \\ (iii) \quad 180 \\ \\\text {Sol. (i) The number 13 can be written as } \\ \text { 11+2 = 13 or 3+3+7 = 13 which is the sum of two or more primes.} \\ \\\text {(ii) The number 130 can be written as } 59+71 =130 \\ \text { which is the sum of two primes.} \\ \\\text {(iii) The number 180 can be written as 79+101 or 139+17+11+13 } \\ \text { which is the sum of two primes.} \\ \\ \end{array}