Ex 3.1
Ques 1: Write all the factors of the following numbers:
(a) 24
(b) 15
(c) 21
(d) 27
(e) 12
(f) 20
(g) 18
(h) 23
(i) 36
Ans:
(a) 24 = 1 x 24 = 2 x 12 = 3 x 8 = 4 x 6 = 6 x 4
âˆ´ Factors of 24 = 1, 2, 3, 4, 6, 12, 24
(b) 15 = 1 x 15 = 3 x 5 = 5 x 3
âˆ´ Factors of 15 = 1, 3, 5, 15
(c) 21 = 1 x 21 = 3 x 7 = 7 x 3
âˆ´ Factors of 21 = 1, 3, 7, 21
(d) 27 = 1 x 27 = 3 x 9 = 9 x 3
âˆ´ Factors of 27 = 1, 3, 9, 27
(e) 12 = 1 x 12 = 2 x 6 = 3 x 4 = 4 x 3
âˆ´ Factors of 12 = 1, 2, 3, 4, 6, 12
(f) 20 = 1 x 20 = 2 x 10 = 4 x 5 = 5 x 4
âˆ´ Factors of 20 = 1, 2, 4, 5, 10, 20
(g) 18 = 1 x 18 = 2 x 9 = 3 x 6
âˆ´ Factors of 18 = 1, 2, 3, 6, 9, 18
(h) 23 = 1 x 23
âˆ´ Factors of 23 = 1, 23
(i) 36 = 1 x 36 = 2 x 18 = 3 x 12 = 4 x 9 = 6 x 6
âˆ´ Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
Ques 2: Write first five multiplies of:
(a) 5
(b) 8
(c) 9
Ans: (a) 5 x 1 = 5, 5 x 2 = 10, 5 x 3 = 15, 5 x 4 = 20, 5 x 5 = 25
âˆ´ First five multiples of 5 are 5, 10, 15, 20, 25.
(b) 8 x 1 = 8, 8 x 2 = 16, 8 x 3 = 24, 8 x 4 = 32, 8 x 5 = 40
âˆ´ First five multiples of 8 are 8, 16, 24, 32, 40.
(c) 9 x 1 = 9, 9 x 2 = 18, 9 x 3 = ,27, 9 x 4 = 36, 9 x 5 = 45
âˆ´ First five multiples of 9 are 9, 18, 27, 36, 45.
Ques 3: Match the items in column 1 with the items in column 2:
Ans: (i) â†’ (b)
(ii) â†’ (d)
(iii) â†’ (a)
(iv) â†’ (f)
(v) â†’ (e)
Ques 4: Find all the multiples of 9 up to 100.
Ans: Multiples of 9 up to 100 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99
Ex 3.2
Ques 1: What is the sum of any two
(a) Odd numbers?
(b) Even numbers?
Ans: (a) The sum of any two odd numbers is an even number.
Example: 1 + 3 = 4, 3 + 5 = 8
(b) The sum of any two even numbers is also an even number.
Example: 2 + 4 = 6, 6 + 8 = 14
Ques 2: State whether the following statements are True or False:
(a) The sum of three odd numbers is even.
(b) The sum of two odd numbers and one even number is even.
(c) The product of three odd numbers is odd.
(d) If an even number is divided by 2, the quotient is always odd.
(e) All prime numbers are odd.
(f) Prime numbers do not have any factors.
(g) Sum of two prime numbers is always even.
(h) 2 is the only even prime number.
(i) All even numbers are composite numbers.
(j) The product of two even numbers is always even.
Ans: (a) False
(b) True
(c) True
(d) False
(e) False
(f) False
(g) False
(h) True
(i) False
(j) True
Ques 3: The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 100.
Ans: 17 and 71; 37 and 73; 79 and 97
Ques 4: Write down separately the prime and composite numbers less than 20.
Ans: Prime numbers : 2, 3, 5, 7, 11, 13, 17, 19
Composite numbers : 4, 6, 8, 9, 10, 12, 14, 15, 16, 18
Ques 5: What is the greatest prime number between 1 and 10?
Ans: The greatest prime number between 1 and 10 is â€˜7â€™.
Ques 6: Express the following as the sum of two odd primes.
(a) 44
(b) 36
(c) 24
(d) 18
Ans: (a) 3 + 41 = 44
(b) 5 + 31 = 36
(c) 7 + 17 = 24
(d) 7 + 11 = 18
Ques 7: Give three pairs of prime numbers whose difference is 2.
[Remark: Two prime numbers whose difference is 2 are called twin primes].
Ans: 3 and 5;
5 and 7;
11 and 13
Ques 8: Which of the following numbers are prime?
(a) 23
(b) 51
(c) 37
(d) 26
Ans: (a) 23 and (c) 37 are prime numbers.
Ques 9: Write seven consecutive composite numbers less than 100 so that there is no prime number between them.
Ans: Seven consecutive composite numbers: 90, 91, 92, 93, 94, 95, 96
Ques 10: Express each of the following numbers as the sum of three odd primes:
(a) 21
(b) 31
(c) 53
(d) 61
Ans: (a) 21 = 3 + 7 + 11
(b) 31 = 3 + 11 + 17
(c) 53 = 13 + 17 + 23
(d) 61 = 19 + 29 + 13
Ques 11: Write five pairs of prime numbers less than 20 whose sum is divisible by 5.
(Hint: 3 + 7 = 10)
Ans: 2 + 3 = 5;
7 + 13 = 20;
3 + 17 = 20;
2 + 13 = 15;
5 + 5 = 10
Ques 12: Fill in the blanks:
(a) A number which has only two factors is called a _______.
(b) A number which has more than two factors is called a _______.
(c) 1 is neither _______ nor _______.
(d) The smallest prime number is _______.
(e) The smallest composite number is _______.
(f) The smallest even number is _______.
Ans: (a) Prime number
(b) Composite number
(c) Prime number and composite number
(d) 2
(e) 4
(f) 2
Ex 3.3
Ques 1: Using divisibility tests, determine which of the following numbers are divisible by 2; by 3; by 4; by 5; by 6; by 8; by 9; by 10; by 11 (say, yes or no):
Number | Divisible by | ||||||||
| 2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 11 |
128 | Yes | No | Yes | No | No | Yes | No | No | No |
990 | |||||||||
1586 | |||||||||
275 | |||||||||
6686 | |||||||||
639210 | |||||||||
429714 | |||||||||
2856 | |||||||||
3060 | |||||||||
406839 |
Answer:
Number | Divisible by | ||||||||
2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 11 | |
128 | Yes | No | Yes | No | No | Yes | No | No | No |
990 | Yes | Yes | No | Yes | Yes | No | Yes | Yes | Yes |
1586 | Yes | No | No | No | No | No | No | No | No |
275 | No | No | No | Yes | No | No | No | No | Yes |
6686 | Yes | No | No | No | No | No | No | No | No |
639210 | Yes | Yes | No | Yes | Yes | No | No | Yes | Yes |
429714 | Yes | Yes | No | No | Yes | No | Yes | No | No |
2856 | Yes | Yes | Yes | No | Yes | Yes | No | No | No |
3060 | Yes | Yes | Yes | Yes | Yes | No | Yes | Yes | No |
406839 | No | Yes | No | No | No | no | No | No | No |
Ques 2: Using divisibility tests, determine which of the following numbers are divisible by 2; by 3; by 4; by 5; by 6; by 8; by 9; by 10; by 11 (say, yes or no):
Using divisibility tests, determine which of the following numbers are divisible by 4; by 8:
(a) 572
(b) 726352
(c) 5500
(d) 6000
(e) 12159
(f) 14560
(g) 21084
(h) 31795072
(i) 1700
(j) 2150
Ans: (a) 572
â†’ Divisible by 4 as its last two digits are divisible by 4.
â†’ Not divisible by 8 as its last three digits are not divisible by 8.
(b) 726352
â†’ Divisible by 4 as its last two digits are divisible by 4.
â†’ Divisible by 8 as its last three digits are divisible by 8.
(c) 5500
â†’ Divisible by 4 as its last two digits are divisible by 4.
â†’ Not divisible by 8 as its last three digits are not divisible by 8.
(d) 6000
â†’ Divisible by 4 as its last two digits are 0.
â†’ Divisible by 8 as its last three digits are 0.
(e) 12159
â†’ Not divisible by 4 and 8 as it is an odd number.
(f) 14560
â†’ Divisible by 4 as its last two digits are divisible by 4.
â†’ Divisible by 8 as its last three digits are divisible by 8.
(g) 21084
â†’ Divisible by 4 as its last two digits are divisible by 4.
â†’ Not divisible by 8 as its last three digits are not divisible by 8.
(h) 31795072
â†’ Divisible by 4 as its last two digits are divisible by 4.
â†’ Divisible by 8 as its last three digits are divisible by 8.
(i) 1700
â†’ Divisible by 4 as its last two digits are 0.
â†’ Not divisible by 8 as its last three digits are not divisible by 8.
(j) 5500
â†’ Not divisible by 4 as its last two digits are not divisible by 4.
â†’ Not divisible by 8 as its last three digits are not divisible by 8.
Ques 3: Using divisibility test, determine which of the following numbers are divisible by 6:
(a) 297144
(b) 1258
(c) 4335
(d) 61233
(e) 901352
(f) 438750
(g) 1790184
(h) 12583
(i) 639210
(j) 17852
Ans: (a) 297144
â†’ Divisible by 2 as its units place is an even number.
â†’ Divisible by 3 as sum of its digits (= 27) is divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is also divisible by 6.
(b) 1258
â†’ Divisible by 2 as its units place is an even number.
â†’ Not divisible by 3 as sum of its digits (= 16) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(c) 4335
â†’ Not divisible by 2 as its units place is not an even number.
â†’ Divisible by 3 as sum of its digits (= 15) is divisible by 3.
Since the number is not divisible by 2 , therefore, it is not divisible by 6.
(d) 61233
â†’ Not divisible by 2 as its units place is not an even number.
â†’ Divisible by 3 as sum of its digits (= 15) is divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(e) 901352
â†’ Divisible by 2 as its units place is an even number.
â†’ Not divisible by 3 as sum of its digits (= 20) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(f) 438750
â†’ Divisible by 2 as its units place is an even number.
â†’ Divisible by 3 as sum of its digits (= 27) is not divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.
(g) 1790184
â†’ Divisible by 2 as its units place is an even number.
â†’ Divisible by 3 as sum of its digits (= 30) is not divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.
(h) 12583
â†’ Not divisible by 2 as its units place is not an even number.
â†’ Not divisible by 3 as sum of its digits (= 19) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(i) 639210
â†’ Divisible by 2 as its units place is an even number.
â†’ Divisible by 3 as sum of its digits (= 21) is not divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.
(j) 17852
â†’ Divisible by 2 as its units place is an even number.
â†’ Not divisible by 3 as sum of its digits (= 23) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
Ques 4: Using divisibility test, determine which of the following numbers are divisible by 11:
(a) 5445
(b) 10824
(c) 7138965
(d) 70169308
(e) 10000001
(f) 901153
Ans:
(a) 5445
â†’ Sum of the digits at odd places = 4 + 5 = 9
â†’ Sum of the digits at even places = 4 + 5 = 9
â†’ Difference of both sums = 9 â€“ 9 = 0
Since the difference is 0, therefore, the number is divisible by 11.
(b) 10824
â†’ Sum of the digits at odd places = 4 + 8 +1 = 13
â†’ Sum of the digits at even places = 2 + 0 = 2
â†’ Difference of both sums = 13 â€“ 2 = 11
Since the difference is 11, therefore, the number is divisible by 11.
(c) 7138965
â†’ Sum of the digits at odd places = 5 + 9 + 3 + 7 = 24
â†’ Sum of the digits at even places = 6 + 8 + 1 = 15
â†’ Difference of both sums = 24 â€“ 15 = 9
Since the difference is neither 0 nor 11, therefore, the number is not divisible by 11.
(d) 70169308
â†’ Sum of the digits at odd places = 8 + 3 + 6 + 0 = 17
â†’ Sum of the digits at even places = 0 + 9 + 1 + 7 = 17
â†’ Difference of both sums = 17 â€“ 17 = 0
Since the difference is 0, therefore, the number is divisible by 11.
(e) 10000001
â†’ Sum of the digits at odd places = 1 + 0 + 0 + 0 = 1
â†’ Sum of the digits at even places = 0 + 0 + 0 + 1 = 1
â†’ Difference of both sums = 1 â€“ 1 = 0
Since the difference is 0, therefore, the number is divisible by 11.
(f) 901153
â†’ Sum of the digits at odd places = 3 + 1 + 0 = 4
â†’ Sum of the digits at even places = 5 + 1 + 9 = 15
â†’ Difference of both sums = 15 â€“ 4 = 11
Since the difference is 11, therefore, the number is divisible by 11.
Ques 5: Write the smallest digit and the largest digit in the blanks space of each of the following numbers so that the number formed is divisibly by 3:
(a) __________ 6724
(b) 4765 __________ 2
Ans: (a) We know that a number is divisible by 3 if the sum of all digits is divisible by 3.
Therefore, Smallest digit : 2 â†’ 26724 = 2 + 6 + 7 + 2 + 4 = 21
Largest digit: 8 â†’ 86724 = 8 + 6 + 7 + 2 + 4 = 27
(b) We know that a number is divisible by 3 if the sum of all digits is divisible by 3.
Therefore,
Smallest digit: 0 â†’ 476502 = 4 + 7 + 6 + 5 + 0 + 2 = 24
Largest digit: 9 â†’ 476592 = 4 + 7 + 6 + 5 + 0 + 2 = 33
Ques 6: Write the smallest digit and the largest digit in the blanks space of each of the following numbers so that the number formed is divisibly by 11:
(a) 92 __________ 389
(b) 8 __________ 9484
Ans: (a) We know that a number is divisible by 11 if the difference of the sum of the digits at odd places and that of even places should be either 0 or 11.
Therefore, 928389 â†’
Odd places = 9 + 8 + 8 = 25
Even places = 2 + 3 + 9 = 14
Difference = 25 â€“ 14 = 11
(b) We know that a number is divisible by 11 if the difference of the sum of the digits at odd places and that of even places should be either 0 or 11.
Therefore, 869484 â†’
Odd places = 8 + 9 + 8 = 25
Even places = 6 + 4 + 4 = 14
Difference = 25 â€“ 14 = 11