EXERCISE-2A
Question 1
Q.1. Define (i) factor (ii) multiple. Give five
examples of each.
Ans.
(i) A factor of a number is an exact divisor of that number.
Examples:
1. 2 is a factor of 8
2. 5 is a factor of 15
3. 9 is a factor of 27
4. 4 is a factor of 20
5. 3 is a factor of 12.
(ii) Multiple. A number is said to be a
multiple of any of its factors.
Examples:
1. 15 is a multiple of 3
2. 8 is a multiple of 4
3. 10 is a multiple of 2
4. 25 is a multiple of 5
5. 18 is a multiple of 9 .
Question 2
Q.2. Write down all the factors of :
(i) 20
(ii) 36
(iii) 60
(iv) 75
Ans.
(i) We know that
$20=1 \times 20,20=2 \times 10,20=4 \times 5$
which shows that the numbers $1,2,4$,
$5,10,20$ exactly divide 20.
$\therefore 1,2,4,5,10$ and 20 are all factors of 20 .
(ii) We know that
$ \begin{aligned} & 36=1 \times 36,36=2 \times 18,36=3 \times 12 \\ & 36=4 \times 9,36=6 \times 6 \end{aligned} $
This shows that each of the numbers 1 ,
$2,3,4,6,9,12,18,36$ exactly divides 36 .
$\therefore 1,2,3,4,6,9,12,18,36$ are the factors of 36 .
(iii) We know that
$ \begin{aligned} & 60=1 \times 60,60=2 \times 30,60=3 \times 20, \\ & 60=4 \times 15,60=5 \times 12,60=6 \times 10 \end{aligned} $
This shows that each of the numbers $1,2,3,4,5,6,10,12,15,20,30,60$
exactly divides 60 .
$\therefore 1,2,3,4,5,6,10,12,15,20,30,60$ are all the factors of 60 .
(iv) We know that
$75=1 \times 75,75=3 \times 25,75=5 \times 15$
This shows that each of the numbers 1,3,5,15,25,75 exactly divides 75 . $\therefore 1,3,5,15,25,75$ are all the factors of 75 .
Question 3
Q.3. Write the first five multiples of each of the following numbers:
(i) 17
(ii) 23
(iii) 65
(iv) 70
Ans.
(i) First five multiples of 17 are :
$ \begin{aligned} & 17 \times 1=17 \\ & 17 \times 2=34 \\ & 17 \times 3=51 \\ & 17 \times 4=68 \\ & 17 \times 5=85 \end{aligned} $
(ii) First five multiples of 23 are :
$
\begin{aligned}
& 23 \times 1=23 \\
& 23 \times 2=46 \\
& 23 \times 3=69 \\
& 23 \times 4=92 \\
& 23 \times 5=115
\end{aligned}
$
(iii) First five multiples of 65 are :
$
\begin{aligned}
& 65 \times 1=65 \\
& 65 \times 2=130 \\
& 65 \times 3=195 \\
& 65 \times 4=260 \\
& 65 \times 5=325
\end{aligned}
$
(iv) First five multiples of 70 are :
$
\begin{aligned}
& 70 \times 1=70 \\
& 70 \times 2=140 \\
& 70 \times 3=210 \\
& 70 \times 4=280 \\
& 70 \times 5=350
\end{aligned}
$
Question 4
Q.4. Which of the following numbers are even and which are odd.
(i) 32
(ii) 37
(iii) 50
(iv) 58
(v) 69
(vi) 144
(vii) 321
(viii) 253
Ans.
(i) 32 is a multiple of 2 , so it is an even number.
(ii) 37 is not a multiple of 2 , so it is an odd number.
(iii) 50 is a multiple of 2 , so it is an even number.
(iv) 58 is a multiple of 2 , so it is an even number.
(v) 69 is not a multiple of 2 , so it is an odd number.
(vi) 144 is a multiple of 2 , so it is an even number.
(vii) 321 is not a multiple of 2 , so it is an odd number.
(viii) 253 is not a multiple of 2 , so it is an odd number.
Question 5
Ans.
Prime Numbers. Each of the numbers which has exactly two factors, namely 1 and itself, is called a prime number.
Examples. The numbers $2,3,5,7,11,13,17,19,23,29$ are all prime numbers.
Question 6
(i) 10 and 40
(ii) 80 and 100
(iii) 40 and 80
(iv) 30 and 40
Sol.
(i) Prime numbers between 10 and 40 are :
11,13,17,19,23,29,31,37 .
(ii) Prime numbers between 80 and 100 are :
(ii) Prime numbers between 80 and 100 are :
83, 89, 97
(iii) Prime numbers between 40 and 80 are :
41,43,47,53,59,61,67,71,73,79
(iv) Prime numbers between 30 and 40 are :
(iii) Prime numbers between 40 and 80 are :
41,43,47,53,59,61,67,71,73,79
(iv) Prime numbers between 30 and 40 are :
31, 37
Question 7
Q. 7.
Question 7
Q. 7.
(i) Write the smallest prime number.
(ii) List all even prime numbers.
(iii) Write the smallest odd prime number.
Ans.
(ii) List all even prime numbers.
(iii) Write the smallest odd prime number.
Ans.
(i) 2 is the smallest prime number.
(ii) 2 is the only even prime number.
(iii) 3 is the smallest odd prime number.
Question 8
Q.8. Find which of the following numbers are prime :
(i) 87
(ii) 89
(iii) 63
(iv) 91
(ii) 2 is the only even prime number.
(iii) 3 is the smallest odd prime number.
Question 8
(i) 87
(ii) 89
(iii) 63
(iv) 91
Ans :
This shows that 1,3,29,87 are the factors of 87.
$\therefore$ The number 87 is not a prime number as it has more than 2 factors.
(i) We know that
$87=1 \times 87,87=3 \times 29$This shows that 1,3,29,87 are the factors of 87.
$\therefore$ The number 87 is not a prime number as it has more than 2 factors.
(ii) We have $89=1 \times 89$
$\therefore$ The number 89 is a prime number as it has only 2 factors.
$\therefore$ The number 89 is a prime number as it has only 2 factors.
(iii) We have $63=1 \times 63,63=3 \times 21$,
$63=7 \times 9$
This shows that the number 63 has more than 2 factors namely $1,3,7,9,21,63$. So, it is not a prime number.
$63=7 \times 9$
This shows that the number 63 has more than 2 factors namely $1,3,7,9,21,63$. So, it is not a prime number.
(iv) We have $91=1 \times 91,91=7 \times 13$
This shows that the number 91 has more than 2 factors namely $1,7,13,91$. So, it is not a prime number.
Question 9
Q.9. Make a list of seven consecutive numbers, none of which is prime.
Ans.
From the Sieve of Eratosthenes, we see that the seven consecutive numbers are $90,91,92,93,94,95$ and 96 .
Question 10
Q.10.
Question 10
Q.10.
(i) Is there any counting number having no factor at all ?
(ii) Find all the numbers having exactly one factor.
(iii) Find numbers between 1 and 100 having exactly three factors.
Ans.
(ii) Find all the numbers having exactly one factor.
(iii) Find numbers between 1 and 100 having exactly three factors.
Ans.
(i) There is no counting number having no factor at all.
(ii) The number 1 has exactly one factor.
(iii) The numbers between 1 and 100 having exactly three factors are : 4, 9, 25, 49.
Question 11
Q.11. What are composite numbers ? Can a composite number be odd ? If yes, write the smallest odd composite number.
Ans.
(ii) The number 1 has exactly one factor.
(iii) The numbers between 1 and 100 having exactly three factors are : 4, 9, 25, 49.
Question 11
Q.11. What are composite numbers ? Can a composite number be odd ? If yes, write the smallest odd composite number.
Ans.
Composite Numbers. Numbers having more than two factors are called composite numbers. A composite number can be an odd number. The smallest odd composite number is 9 .
Question 12
Ans.
Twin-primes. Two consecutive odd prime numbers are known as twin primes.
The prime numbers between 50 and 100 are:
53,59,61,67,71,73,79,83,89,97
From above pairs of twin-primes are
(59,61),(71,73)
Question 13
Q.13. What are co-primes? Give examples of five pairs of co-primes. Are co-primes always prime ? If no, illustrate your answer by an example.
Ans.
Co-primes. Two numbers are said to be co-prime if they do not have a common factor.
Examples. Five pairs of co-primes are:
(i) 2,3
(ii) 3,4
(iii) 4,5
(iv) 8,15
(v) 9,16
Co-primes are not always prime.
Illustration. In the pair $(3,4)$ of coprimes, 3 is a prime number whereas 4 is a composite number.
Question 14
Q.14. Express each of the following numbers as the sum of two odd primes:
(i) 36
(ii) 42
(iii) 84
(iv) 98
Sol.
(i) $36=7+29$
(ii) $42=5+37$
(iii) $84=17+67$
(iv) $98=19+79$
Question 15
Q.15. Express each of the following odd numbers as the sum of three odd prime numbers:
(i) 31
(ii) 35
(iii) 49
(iv) 63
Ans.
Question 17
Q.17. Which of the following statements are true ?
(i) 1 is the smallest prime number.
(ii) If a number is prime, it must be odd.
(iii) The sum of two prime numbers is always a prime number.
(iv) If two numbers are co-prime, at least one of them must be a prime number.
Ans.
(i) to (iv). None of the given statements is true.
TESTS FOR DIVISIBILITY OF NUMBERS
(i) Test of divisibility by 10. A number is divisible by 10 , if its unit's digit is zero.
(ii) Test of divisibility by 5 . A number is divisible by 5 , if its unit's digit is 0 or 5 .
(iii) Test of divisibility by 2. A number is divisible by 2 , if its unit's digit is $0,2,4,6$ or 8 .
(iv) Test of divisibility by 3. A number is divisible by 3 , if the sum of its digits is divisible by 3 .
(v) Test of divisibility by 9 . A number is divisible by 9 , if the sum of its digits is divisible by 9 .
(vi) Test of divisibility by 4. A number is divisible by 4 , if the number formed by its digits in ten's and unit's places is divisible by 4 .
GENERAL PROPERTIES OF DIVISIBILITY
Property 1. If a number is divisible by another number, it must be divisible by each of the factors of that number.
Property 2. If a number is divisible by each of the two co-prime numbers, it must be divisible by their product.
Property 3. If a number is a factor of each of the two given numbers then it must be a factor of their sum.
Property 4. If a number is a factor of each of the two given numbers then it must be a factor of their difference.
The prime numbers between 50 and 100 are:
53,59,61,67,71,73,79,83,89,97
From above pairs of twin-primes are
(59,61),(71,73)
Question 13
Q.13. What are co-primes? Give examples of five pairs of co-primes. Are co-primes always prime ? If no, illustrate your answer by an example.
Ans.
Co-primes. Two numbers are said to be co-prime if they do not have a common factor.
Examples. Five pairs of co-primes are:
(i) 2,3
(ii) 3,4
(iii) 4,5
(iv) 8,15
(v) 9,16
Co-primes are not always prime.
Illustration. In the pair $(3,4)$ of coprimes, 3 is a prime number whereas 4 is a composite number.
Question 14
(i) 36
(ii) 42
(iii) 84
(iv) 98
Sol.
(i) $36=7+29$
(ii) $42=5+37$
(iii) $84=17+67$
(iv) $98=19+79$
Question 15
Q.15. Express each of the following odd numbers as the sum of three odd prime numbers:
(i) 31
(ii) 35
(iii) 49
(iv) 63
Ans.
(i) $31=5+7+19$
(ii) $35=5+7+23$
(iii) $49=3+5+41$
(iv) $63=7+13+43$
Question 16
Q.16. Express each of the following numbers as the sum of twin primes:
(i) 36
(ii) 84
(iii) 120
(iv) 144
Ans.
(i) $36=17+19$
(ii) $84=41+43$
(iii) $120=59+61$
(iv) $144=71+73$
(ii) $35=5+7+23$
(iii) $49=3+5+41$
(iv) $63=7+13+43$
Question 16
(i) 36
(ii) 84
(iii) 120
(iv) 144
Ans.
(i) $36=17+19$
(ii) $84=41+43$
(iii) $120=59+61$
(iv) $144=71+73$
Question 17
(i) 1 is the smallest prime number.
(ii) If a number is prime, it must be odd.
(iii) The sum of two prime numbers is always a prime number.
(iv) If two numbers are co-prime, at least one of them must be a prime number.
Ans.
(i) to (iv). None of the given statements is true.
TESTS FOR DIVISIBILITY OF NUMBERS
(i) Test of divisibility by 10. A number is divisible by 10 , if its unit's digit is zero.
(ii) Test of divisibility by 5 . A number is divisible by 5 , if its unit's digit is 0 or 5 .
(iii) Test of divisibility by 2. A number is divisible by 2 , if its unit's digit is $0,2,4,6$ or 8 .
(iv) Test of divisibility by 3. A number is divisible by 3 , if the sum of its digits is divisible by 3 .
(v) Test of divisibility by 9 . A number is divisible by 9 , if the sum of its digits is divisible by 9 .
(vi) Test of divisibility by 4. A number is divisible by 4 , if the number formed by its digits in ten's and unit's places is divisible by 4 .
GENERAL PROPERTIES OF DIVISIBILITY
Property 1. If a number is divisible by another number, it must be divisible by each of the factors of that number.
Property 2. If a number is divisible by each of the two co-prime numbers, it must be divisible by their product.
Property 3. If a number is a factor of each of the two given numbers then it must be a factor of their sum.
Property 4. If a number is a factor of each of the two given numbers then it must be a factor of their difference.
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