MATHEMATICS ASSIGNMENT
CHAPTER1 [NUMBER SYSTEM] CLASS X
Mathematics assignment for class X chapter 1 Real Numbers with answers and hints for the solution. Extra questions of chapter 1 real number class X
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Math Assignment Chapter 1 Class X : Real Number System
Q1 Define Fundamental theorem of Arithmetic’s, HCF & LCM .
Solution: Fundamental theorem of Arithmetic’s :
Every composite number can be expressed as the product of primes and this factorization is unique, irrespective of their order.
HCF : HCF is the product of common factors with smallest powers.
LCM : LCM is the product of all factors with largest powers.
Q2) Express the following as the product of primes.
a) 156 b) 5005 c) 7429 d) 1008 e) 364 f) 429 g) 288 h) 424 i) 10626
Question 3: Use FTA & Find HCF & LCM
of the following
i) 510 , 92 Ans HCF = 2, LCM = 23460
ii) 336, 54 Ans HCF = 6, LCM = 3024
iii) 225 , 450 Ans HCF = 225, LCM = 450
iv) 96 and 120. Ans: HCF = 24, LCM = 480v) 1440 , 1800 Ans HCF = 360, LCM = 7200
vi) 18180
and 7575 Ans HCF = 1515, LCM = 90900
vii) 40, 36 , 126 Ans HCF = 2, LCM = 2520
viii) 25, 40 , 60 Ans HCF = 5, LCM = 600
ix) 72, 80, 120 Ans HCF = 8, LCM = 720
x) 84, 90 , 120 Ans HCF = 6, LCM = 2520
xi) 28, 44, 132 Ans HCF = 4 LCM = 924
Question 4: If the product of two
coprime numbers is 550, then their HCF is : Ans: 1
Question 5 If two positive integers
p and q can be expressed as p = 18a^{2}b^{4} and q = 20a^{3}b^{2},
where a and b are prime numbers, then find LCM(p, q) Ans : LCM = 180 a^{3}b^{4} HCF = 2a^{2}b^{2}
Question 6: If ‘p’ and ‘q’ are
natural numbers and ‘p’ is the multiple of ‘q’, then what is the HCF of ‘p’ and
‘q’ ? Ans HCF = q, LCM = p
Question 7: Given that LCM(989, 1892) is 43516, find HCF. Ans 43
Question 8: HCF (45 and 105) = 15 find their LCM. Ans: 315
Question 9: Given that LCM(77, 99) = 693. Find HCF. Ans: 11
Question 10: Given that HCF(x, 657) = 9 and LCM(x, 657) = 22338. Find x.
Ans 306
Question 11: If HCF(6, x) = 2 and LCM(6, x) = 60. Find x. Ans 20
Question 12: If HCF(48, x) = 24 and LCM(48, x) = 144, find x. Ans 72
Question 13: Given that HCF (306,
1314) = 18, find LCM of (306, 1314). Ans 22338
Question 14: Prove the following are irrational
number
S N

Questions

S N

Questions

S N

Questions

a


e


i


b


f


j


c


g


k 

d


h


l  
Question 15: If the HCF (2520, 6600)
= 40 and LCM (2520, 6600) = 252 × k, then find the value of k ? Ans: k = 1650
Question 16: a) If
a = 2^{2} × 3^{x} , b = 2^{2} × 3 × 5, c = 2^{2} × 3 × 7 and LCM(a, b, c) = 3780, then find the value of x ? Ans : x = 3
b) Two positive integers m
and n are expressed as m = p^{5}q^{2} and n = p^{3}q^{4},
where p and q are prime numbers. Then find the LCM of m and nAns: p^{5}q^{4}
Question 17: The HCF of two numbers
65 and 104 is 13. If LCM of 65 and 104 is 40x then find the value of x ? Ans: x = 13
Question 18: Find the ratio of HCF to LCM
of the least composite number and the least prime number. Ans: 1 : 2Question 19: Check whether 12ⁿ can end with the digit 0 ∀n ϵ N.
Question 20: Can the number (15)^{n}, n being a natural number, end with the digit 0 ?
Question 21: Two numbers are in the
ratio 2 : 3 and their LCM is 180. What is the HCF of these numbers ?Ans: x = 30,
Required numbers are 60 and 90, HCF = 30
Solution Hint: Let two numbers are 2x and 3x
HCF of 2x and 3x = x
LCM of 2x and 3x = 6x
ATQ 6x = 180 ⇒ x = 30
Question 22: What is the exponent of 3 in the prime factorization of 864.
Ans: In the prime factorisation of 864 the Exponent of 3 is 3
Question 23: What is the HCF × LCM for 105 and 120. Ans: 12600
Question 24: HCF x LCM for the
numbers 40 and 30 is : Ans 1200Question 25: Can two numbers have 15 as their HCF and 175 as their LCM ? Give reasons.
Answer: 175 is not divisible by 15, so it is not possible to have two numbers which have 15 as their HCF and 175 as their LCM
Note: LCM is always the multiple of HCF. So if we divide LCM of two numbers by their HCF then remainder should be zero.
Question 26:
a) Show that 2 × 3 × 4 × 5 × 6 × 7 + 5 × 6 is a composite number.
Hint: Composite Numbers: The numbers which have more than two factors are called composite numbers.
b) Show that 11 × 19 × 23 +
3 × 11 is not a prime numberQuestion 27: Show
that the number 5 × 11 × 17 × 3 + 11 is a composite number.
Question 28: Write whether is a rational or an irrational number. Ans Rational number
Hint: Simplify the above number we get a rational number
Question 29: What is the smallest number which, when divided by 35, 56, 91 leaves remainder 7 in each case. Ans 3647
Question 30: Find the greatest number that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively. Ans 63
Question 31:
Find the greatest number
which divides 281 and 1249, leaving remainder 5 and 7 ? Ans : 138
Question 32: Find the greatest number which on dividing 1657 and 2037 leaves remainder 6 and 5 respectively. Ans 127
Question 33: Find the greatest number of 6 digit which is exactly divisible by 24, 15 and 36
Question 34: Length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm respectively. Find the length of the longest rod which can measure the three dimensions of the room exactly. Ans: 75 cm
Question 35: Find the greatest number which when divides 2011 and 2623 leaving remainder 9 and 5 respectively. Ans 154
Question 36: Find the largest number which on dividing 1251, 9377 and 15628 leaves remainder 1, 2, and 3 respectively. Ans 625
Question 37
On a morning walk, three persons step out together and their steps measure 30 cm, 36 cm and 40 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps. Ans 360
Question 38Three bells ring at
intervals of 6, 12 and 18 minutes. If all the three bells rang at 6 a. m., when
will they ring together again ?
Ans: LCM = 36
minutes, Three bells rang again at 6:36 A.M.
Question 39The traffic lights at
three different road crossings change after every 48 seconds, 72 seconds and
108 seconds respectively. If they change simultaneously at 7 a.m., at what time
will they change together next ?
Ans: LCM = 432
second = 7minutes 12 seconds.Three bells rang
again at 7h : 07 min : 12 sec
Question 40 (Case Study Based Question)Teaching mathematics through activities is a
powerful approach that enhances students understanding and engagement. Keeping
this in mind, Ms. Mukta planned a prime number game for class 5 students. She
announces the number 2 in her class and asked the first student to multiply it
by a prime number and then pass it to the next student. Second student also
multiplied it by a prime number and passed it to third student. In this way by
multiplying to a prime number, the last student got 173250
Now,
Mukta asked some questions as given below to the students:
(i)
What is the least prime number used by students ?
(ii) (a) How many students are in the class ?
OR
(b) What is the highest prime number
used by students ?
(iii)
Which prime number has been used maximum times ?
(i)Ans: 3 (ii)a Ans: 7 Students (ii) b Ans: 11 (iii) Ans: 5
Question 41In a teacher’s workshop, the number of teachers teaching
French, Hindi and English are 48, 80 and 144 respectively. Find the minimum
number of rooms required if in each room the same number of teachers are seated
and all of them are of the same subject.
SolutionMinimum number of rooms required means there should be
maximum number of teachers in a room. We have to find HCF of 48, 80 and 144.
48 = 2^{4} × 3
80 = 2^{4} × 5
144 = 2^{4} × 3^{2}
HCF(48, 80, 144) = 2^{4} = 16
Therefore, total number of rooms required:
QUESTIONS DELETED FROM CBSE SYLLABUS
* Q1 State whether the following are terminating or nonterminating, if terminating then convert these into decimals.
a) Ans Terminating 0.45
b) Ans nonterminating but repeating decimal
c) Ans Terminating
d) Ans Terminating
e) Ans Terminating
f) Ans Terminating 40.6
* Question 2
Show that any positive even integer is of the form 4q or 4q + 2.
* Question 3
Show that any positive even integers is of the form 6q, 6q + 2, 6q + 4.
* Question 4
Use E.D.A to show that square of any positive integer is either of the form 3m, 3m + 1 for some integer m.
* Question 5
Use E.D.A to show that cube of any positive integer is either of the form 9m, 9m + 1, 9m + 8.
* Question 6
If HCF of 144 and 180 is expressed in the form 13m – 3 , then find the value of m.
Ans 3
* Question 7
If HCF of 65 and 117 is expressible in the form 65n  117, then find the value of n.
Ans n = 2
* Question 8
Find the HCF of 65 and 117 and find the value of m and n if HCF = 65m + 117n
Ans m = 2, n =  1
* Question 9
By Using EDA, find whether 847, 2160 are coprimes or not
Ans Yes these are coprime because their HCF = 1
* Question 10
Given The decimal expansion of above given number will terminate after how many places of decimals.
Ans 4
* Question 11
If HCF of 81 and 237 be expressed as 81x + 237y then find the value of x and y
Ans x =  38, y = 13
* Question 12
Show that one and only one out of n, n + 4, n + 8, n + 12, n + 16 is divisible by 5. Where n is any positive integer.
* Question 13
Find the smallest number which leaves remainder 8 and 12 when divided by 28 and 32 respectively . Ans 204
[Hint 28 – 8 = 20, 32 – 12 = 20, Required No = LCM(28, 32) 20]
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