Table of Contents
Exercise Ex. 1A
Question 1
What do you mean by Euclid’s division lemma?Solution 1
For any two given positive integers a and b there exist unique whole numbers q and r such that
Here, we call ‘a’ as dividend, b as divisor, q as quotient and r as remainder.
Dividend = (divisor quotient) + remainderQuestion 2
A number when divided by 61 gives 27 as quotient and 32 as remainder. Find the number.Solution 2
By Euclid’s Division algorithm we have:
Dividend = (divisor × quotient) + remainder
= (61 27) + 32 = 1647 + 32 = 1679Question 3
By what number should 1365 be divided to get 31 as quotient and 32 as remainder?Solution 3
By Euclid’s Division Algorithm, we have:
Dividend = (divisor quotient) + remainder
Question 4
Using Euclid’s algorithm, find the HCF of:
(i) 405 and 2520
(ii) 504 and 1188
(iii) 960 and 1575Solution 4
(i)
On dividing 2520 by 405, we get
Quotient = 6, remainder = 90
2520 = (405 6) + 90
Dividing 405 by 90, we get
Quotient = 4,
Remainder = 45
405 = 90 4 + 45
Dividing 90 by 45
Quotient = 2, remainder = 0
90 = 45 2
H.C.F. of 405 and 2520 is 45
(ii) Dividing 1188 by 504, we get
Quotient = 2, remainder = 180
1188 = 504 2+ 180
Dividing 504 by 180
Quotient = 2, remainder = 144
504 = 180 × 2 + 144
Dividing 180 by 144, we get
Quotient = 1, remainder = 36
Dividing 144 by 36
Quotient = 4, remainder = 0
H.C.F. of 1188 and 504 is 36
(iii) Dividing 1575 by 960, we get
Quotient = 1, remainder = 615
1575 = 960 × 1 + 615
Dividing 960 by 615, we get
Quotient = 1, remainder = 345
960 = 615 × 1 + 345
Dividing 615 by 345
Quotient = 1, remainder = 270
615 = 345 × 1 + 270
Dividing 345 by 270, we get
Quotient = 1, remainder = 75
345 = 270 × 1 + 75
Dividing 270 by 75, we get
Quotient = 3, remainder =45
270 = 75 × 3 + 45
Dividing 75 by 45, we get
Quotient = 1, remainder = 30
75 = 45 × 1 + 30
Dividing 45 by 30, we get
Remainder = 15, quotient = 1
45 = 30 × 1 + 15
Dividing 30 by 15, we get
Quotient = 2, remainder = 0
H.C.F. of 1575 and 960 is 15
Question 5
Show that every positive integer is either even or odd.Solution 5
Question 6
Show that any positive odd integer is of the form (6m + 1) or (6m + 3) or (6m + 5), where m is some integer.Solution 6
Question 7
Show that any positive odd integer is of the form (4m + 1) or (4m + 3), where in is some integer.Solution 7
Exercise Ex. 1B
Question 1(i)
Using prime factorization, find the HCF and LCM of:
36, 84
In each case, verify that:
HCF x LCM = product of given numbers.Solution 1(i)
Question 1(ii)
Using prime factorization, find the HCF and LCM of:
23, 31
In each case, verify that:
HCF x LCM = product of given numbers.Solution 1(ii)
Question 1(iii)
Using prime factorization, find the HCF and LCM of:
96, 404
In each case, verify that:
HCF x LCM = product of given numbers.Solution 1(iii)
Question 1(iv)
Using prime factorization, find the HCF and LCM of:
144,198
In each case, verify that:
HCF x LCM = product of given numbers.Solution 1(iv)
Question 1(v)
Using prime factorization, find the HCF and LCM of:
396, 1080
In each case, verify that:
HCF x LCM = product of given numbers.Solution 1(v)
Question 1(vi)
Using prime factorization, find the HCF and LCM of:
1152, 1664
In each case, verify that:
HCF x LCM = product of given numbers.Solution 1(vi)
Question 5(i)
Using prime factorization, find the HCF and LCM of:
8, 9, 25Solution 5(i)
Question 5(ii)
Using prime factorization, find the HCF and LCM of:
12, 15, 21Solution 5(ii)
Question 5(iii)
Using prime factorization, find the HCF and LCM of:
17, 23, 29Solution 5(iii)
Question 5(v)
Using prime factorization, find the HCF and LCM of:
30, 72, 432Solution 5(v)
Question 5(vi)
Using prime factorization, find the HCF and LCM of:
21, 28, 36, 45Solution 5(vi)
Question 6
Is it possible to have two numbers whose HCF is 18 and LCM is 760? Give reason.Solution 6
Question 7
Find the simplest form of:
(iv)
Solution 7
(iv)
Question 8
The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other.Solution 8
Question 9
The HCF of two numbers is 145 and their LCM is 2175. If one of the numbers is 725, find the other.Solution 9
Question 12
The HCF of two numbers is 18 and their product is 12960. Find their LCM.Solution 12
Question 15
Find the largest number which divides 438 and 606, leaving remainder 6 in each case.Solution 15
Question 16
Find the largest number which divides 320 and 457 leaving remainders 5 and 7 respectively.Solution 16
Subtracting 5 and 7 from 320 and 457 respectively:
320 – 5 = 315,
457 – 7 = 450
Let us now find the HCF of 315 and 405 through prime factorization:
The required number is 45.Question 17
Find the least number which when divided by 35, 56 and 91 leaves the same remainder 7 in each case.Solution 17
Question 18
Find the smallest number which when divided by 28 and 32 leaves remainders 8 and 12 respectively.Solution 18
Question 19
Find the smallest number which when increased by 17 is exactly divisible by both 468 and 520.Solution 19
Question 20
Find the greatest number of four digits which is exactly divisible by 15, 24 and 36.Solution 20
Question 24
Find the missing numbers in the following factorisation:
Solution 24
By going upward
5 11= 55
55 3= 165
1652 = 330
330 2 = 660Question 25
In a seminar, the number of participants in Hindi, English and mathematics are 60, 84 and 108 respectively. Find the minimum number of rooms required, if in each room, the same number of participants are to be seated and all of them being in the same subject.Solution 25
Question 26
Three sets of English, Mathematics and Science books containing 336, 240 and 96 books respectively have to be stacked in such a way that all the books are stored subjectwise and the height of each stack is the same. How many stacks will be there?Solution 26
Let us find the HCF of 336, 240 and 96 through prime factorization:
Each stack of book will contain 48 books
Number of stacks of the same height
Question 27
Three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length. What is the greatest possible length of each plank?Solution 27
The prime factorization of 42, 49 and 63 are:
42 = 2 3 7, 49 = 7 7, 63 = 3 3 7
H.C.F. of 42, 49, 63 is 7
Hence, greatest possible length of each plank = 7 mQuestion 28
Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm and 12 m 95 cm.Solution 28
7 m = 700cm, 3m 85cm = 385 cm
12 m 95 cm = 1295 cm
Let us find the prime factorization of 700, 385 and 1295:
Greatest possible length = 35cmQuestion 29
Find the maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and the same number of pencils.Solution 29
Let us find the prime factorization of 1001 and 910:
1001 = 11 7 13
910 = 2 5 7 13
H.C.F. of 1001 and 910 is 7 13 = 91
Maximum number of students = 91Question 30
Find the least number of square tiles required to pave the ceiling of a room 15 m 17 cm long and 9 m 2 cm broad.Solution 30
Question 31
Three measuring rods are 64 cm, 80 cm and 96 cm in length. Find the least length of cloth that can be measured an exact number of times, using any of the rods.Solution 31
Let us find the LCM of 64, 80 and 96 through prime factorization:
L.C.M of 64, 80 and 96
=
Therefore, the least length of the cloth that can be measured an exact number of times by the rods of 64cm, 80cm and 96cm = 9.6mQuestion 32
An electronic device makes a beep after every 60 seconds. Another device makes a beep after every 62 seconds. They beeped together at 10 a.m. At what time will they beep together at the earliest?Solution 32
Interval of beeping together = LCM (60 seconds, 62 seconds)
The prime factorization of 60 and 62:
60 = 30 2, 62 = 31 2
L.C.M of 60 and 62 is 30 31 2 = 1860 sec = 31min
electronic device will beep after every 31minutes
After 10 a.m., it will beep at 10 hrs 31 minutesQuestion 33
The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8 hours, then at what time will they again change simultaneously?Solution 33
Question 34
Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12 minutes respectively. In 30 hours, how many times do they toll together?Solution 34
Exercise Ex. 1C
Question 1(i)
Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.
Solution 1(i)
Question 1(ii)
Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.
Solution 1(ii)
Question 1(iii)
Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.
Solution 1(iii)
Question 1(iv)
Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.
Solution 1(iv)
Question 1(v)
Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.
Solution 1(v)
Question 1(vi)
Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.
Solution 1(vi)
Question 2(i)
Without actual division, show that each of the following rational number is a nonterminating repeating decimal.
Solution 2(i)
Question 2(ii)
Without actual division, show that each of the following rational number is a nonterminating repeating decimal.
Solution 2(ii)
Question 2(iii)
Without actual division, show that each of the following rational number is a nonterminating repeating decimal.
Solution 2(iii)
Question 2(iv)
Without actual division, show that each of the following rational number is a nonterminating repeating decimal.
Solution 2(iv)
Question 2(v)
Without actual division, show that each of the following rational number is a nonterminating repeating decimal.
Solution 2(v)
Question 2(vi)
Without actual division, show that each of the following rational number is a nonterminating repeating decimal.
Solution 2(vi)
Question 2(vii)
Without actual division, show that each of the following rational number is a nonterminating repeating decimal.
Solution 2(vii)
Question 2(viii)
Without actual division, show that each of the following rational number is a nonterminating repeating decimal.
Solution 2(viii)
Question 3
Express each of the following as a fraction in simplest form:
Solution 3
Exercise Ex. 1D
Question 1
Define (i) rational numbers, (ii) irrational numbers, (iii) real numbers.Solution 1
Question 2
Classify the following numbers as rational or irrational:
Solution 2
Question 3
Prove that each of the following numbers is irrational:
Solution 3
Question 7
Solution 7
Question 8
Solution 8
Question 10
Solution 10
Question 11
Solution 11
Question 12
Prove that is irrational.Solution 12
Question 13
Solution 13
Question 16
(i) Give an example of two irrationals whose sum is rational.
(ii) Give an example of two irrationals whose product is rational.Solution 16
Question 17
State whether the given statement is true or false:
(i) The sum of two rationals is always rational.
(ii) The product of two rationals is always rational.
(iii) The sum of two irrationals is an irrational.
(iv) The product of two irrationals is an irrational.
(v) The sum of a rational and an irrational is irrational.
(vi) The product of a rational and an irrational is irrational.Solution 17
(i) The sum of two rationals is always rational – True
(ii) The product of two rationals is always rational – True
(iii) The sum of two irrationals is an irrational – False
(iv) The product of two irrationals is an irrational – False
(v) The sum of a rational and an irrational is irrational – True
(vi) The product of a rational and an irrational is irrational – True
Exercise Ex. 1E
Question 1
State Euclid’s division lemma.Solution 1
Question 2
State fundamental theorem of arithmetic.Solution 2
Question 3
Express 360 as product of its prime factors.Solution 3
Question 4
If a and b are two prime numbers then find HCF(a, b).Solution 4
Question 5
If a and b are two prime numbers then find LCM(a, b).Solution 5
Question 6
If the product of two numbers is 1050 and their HCF is 25, find their LCM.Solution 6
Question 7
What is a composite number?Solution 7
A whole number that can be divided evenly by numbers other than 1 or itself.Question 8
If a and b are relatively prime then what is their HCF?Solution 8
Question 9
If the rational number has a terminating decimal expansion, what is the condition to be satisfied by b?Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Show that there is no value of n for which (2n x 5n) ends in 5.Solution 12
Question 13
Is it possible to have two numbers whose HCF is 25 and LCM is 520?Solution 13
Question 14
Give an example of two irrationals whose sum is rational.Solution 14
Question 15
Give an example of two irrationals whose product is rational.Solution 15
Question 16
If a and b are relatively prime, what is their LCM?Solution 16
Question 17
The LCM of two numbers is 1200. Show that the HCF of these numbers cannot be 500. Why?Solution 17
Question 18
Express as a rational number in simplest form.Solution 18
Question 19
Express as a rational number in simplest formSolution 19
Question 20
Explain why 0.15015001500015 … is an irrational number.Solution 20
Question 21
Solution 21
Question 22
Write a rational number betweenand 2.Solution 22
Question 23
Explain why is a rational number.Solution 23
Exercise MCQ
Question 1
Which of the following is a pair of co-primes?
(a) (14, 35)
(b) (18, 25)
(c) (31,93)
(d)(32, 62)Solution 1
Question 2
If a = (22×33×54) and b = (23×32×5) then HCF (a, b) = ?
(a) 90
(b) 180
(c) 360
(d)540Solution 2
Question 3
HCF of (23×32×5), (22×33×52) and (24×3×53×7) is
(a) 30
(b) 48
(c) 60
(d)105Solution 3
Question 4
LCM of (23×3×5) and (24×5×7) is
(a) 40
(b) 560
(c) 1120
(d)1680Solution 4
Question 5
The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, what is the other number?
(a) 36
(b) 45
(c) 9
(d)81Solution 5
Question 6
The product of two numbers is 1600 and their HCF is 5. The LCM of the numbers is
(a) 8000
(b) 1600
(c) 320
(d)1605Solution 6
Question 7
What is the largest number that divides each one of 1152 and 1664 exactly?
(a) 32
(b) 64
(c) 128
(d)256Solution 7
Question 8
What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively?
(a) 13
(b) 9
(c) 3
(d)585Solution 8
Question 9
What is the largest number that divides 245 and 1029, leaving remainder 5 in each case?
(a) 15
(b) 16
(c) 9
(d)5Solution 9
Question 10
Solution 10
Question 11
Euclid’s division lemma states that for any positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy
(a) 1 < r < b
(b) 0 < r ≤ b
(c) 0 ≤ r < b
(d)0 < r < bSolution 11
Question 12
A number when divided by 143 leaves 31 as remainder. What will be the remainder when the same number is divided by 13?
(a) 0
(b) 1
(c) 3
(d)5Solution 12
Question 13
Which of the following is an irrational number?
(a)
(b) 3.1416
(c)
(d) 3.141141114 …Solution 13
Question 14
𝜋 is
(a) an integer
(b) a rational number
(c) an irrational number
(d)none of theseSolution 14
Question 15
(a) an integer
(b) a rational number
(c) an irrational number
(d) none of theseSolution 15
Question 16
2.13113111311113… is
(a) an integer
(b) a rational number
(c) an irrational number
(d)none of theseSolution 16
Question 17
The number 3.24636363 … is
(a) an integer
(b) a rational number
(c) an irrational number
(d)none of theseSolution 17
Question 18
Which of the following rational numbers is expressible as a terminating decimal?
Solution 18
Question 19
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) four decimal placesSolution 19
Question 20
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d)four decimal placesSolution 20
Question 21
The number 1.732 is
(a) an irrational number
(b) a rational number
(c) an integer
(d)a whole numberSolution 21
Question 22
a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then, the least prime factor of (a+b) is
(a) 2
(b) 3
(c) 5
(d)8Solution 22
Question 23
(a) a rational number
(b) an irrational number
(c) a terminating decimal
(d)a nonterminating repeating decimalSolution 23
Question 24
(a) a fraction
(b) a rational number
(c) an irrational number
(d)none of theseSolution 24
Question 25
(a) an integer
(b) a rational number
(c) an irrational number
(a) none of theseSolution 25
Question 26
What is the least number that is divisible by all the natural numbers from 1 to 10 (both inclusive)
(a) 100
(b) 1260
(c) 2520
(d) 5040Solution 26
Exercise FA
Question 1
(a) a terminating decimal
(b) a nonterminating, repeating decimal
(c) a nonterminating and nonrepeating decimal
(d)none of theseSolution 1
Question 2
Which of the following has a terminating decimal expansion?
Solution 2
Question 3
On dividing a positive integer n by 9, we get 7 as remainder. What will be the remainder if (3n – 1) is divided by 9?
(a) 1
(b) 2
(c) 3
(d)4Solution 3
Question 4
Solution 4
Question 5
Show that any number of the form 4n, n ∊ N can never end with the digit 0.Solution 5
Question 6
The HCF of two numbers is 27 and their LCM is 162. If one of the number is 81, find the other.Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Which of the following numbers are irrational?
Solution 9
Question 10
Solution 10
Question 11
Find the HCF and LCM of 12, 15, 18, 27.Solution 11
Question 12
Give an example of two irrationals whose sum is rational.Solution 12
Question 13
Give prime factorization of 4620.Solution 13
Question 14
Find the HCF of 1008 and 1080 by prime factorization method.Solution 14
Question 15
Solution 15
Question 16
Find the largest number which divides 546 and 764, leaving remainders 6 and 8 respectively.Solution 16
Question 17
Solution 17
Question 18
Show that every positive odd integer is of the form (4q + 1) or (4q + 3) for some integer q.Solution 18
Question 19
Show that one and only one out of n, (n+2) and (n+4) is divisible by 3, where n is any positive integer.Solution 19
Question 20
Solution 20
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