Math Assignment Ch-1 Class X | Real Numbers

MATHEMATICS  ASSIGNMENT
CHAPTER-1 [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 = 480

v) 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 co-prime numbers is 550, then their HCF is :  Ans: 1

Question 5 If two positive integers p and q can be expressed as p = 18a²b⁴ and q = 20a³b², where a and b are prime numbers, then find LCM(p, q) 

Ans :  LCM = 180 a³b⁴    HCF = 2a²b²      

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 = 2² × 3^x , b = 2² × 3 × 5, c = 2² × 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 = p⁵q² and n = p³q⁴, where p and q are prime numbers. Then find the LCM of m and n
Ans: p⁵q⁴

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 : 2

Question 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 1200

Question 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 number

Question 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

Ans 999720

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 38

Three 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 39

The 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 41

In 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.

Solution

Minimum 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 = 2⁴ × 3

80 = 2⁴ × 5

144 = 2⁴ × 3²

HCF(48, 80, 144) = 2⁴ = 16

Therefore, total number of rooms required:  

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QUESTIONS DELETED FROM CBSE SYLLABUS

* Q1- State whether the following are terminating or non-terminating, if terminating then convert these into decimals.

a)     Ans Terminating  0.45

b)               Ans  non-terminating 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 co-primes or not
Ans Yes these are co-prime 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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