MATHS Solutions For Class 10 MBOSE II Maths Ex 1(c)

 

CHAPTER-1                      REAL NUMBER

EXRCISE-1(C)

 

SHORT ANSWER TYPE QUESTION (SLAB-I)

1. FIND THE PRIME FRACTION OF THE FOLLOWING NUMBERS:

(i). 96.

SOL:



 


 


 


 



24

48



SOL:  

 

  HENCE, 96=25*3.



(ii). 408

SOL:


2

2

2

17

51



 

 

             ∴ HENCE, 408=23*3*17.

 

(iii). 2025

SOL:



3









 



 

   ∴ HENCE, 2025=34*52.

 

    2. EXPRESS 16380 AS PRIME NUMBER.

SOL:    


2







 

 

  HENCE PRIME FACRORS OF 16380=22*32*5*7*13.

 


3 .EXPRESS THE FOLLOWING NUMBERS AS PRODUCT OF THEIR PRIMES:

(i). 49896

SOL:  






 

 


   ∴ HENCE, 49896=23*3*7*11



(ii).897444







SOL:   


 

 

 HENCE, 874944 = 26*32*72*31.






4. FIND HCF AND OF FOLLOWING PAIRS OF THE INTEGERS BY APPLYING THE FUNDAMENTALS THEOREOM OF ARIRTHMETIC.

(i). 336, 54

SOL: 336=2*2*2*2*3*7

          54=2*3*3*3

       HCF(336,54)=2*3=6

       LCM(336,54)=2*2*2*2*3*3*3*7=3024

(ii). 225, 867

SOL: 225=3*3*5*5

          867=3*17*17

          HCF (225,867)=3

          LCM(225,867)=3*3*5*5*17*17

                                       =9*25*289

                                       =65025.

 

5. FIND HCF AND LCM OF THE FOLLOWING PAIRS OF INTEGERS BY PRIME FACTORISATION METHOD AND VERIFY THAT LCM*HCF= PRODUCT OF TWO NUMBERS:

(i). 26, 91

SOL: 26=2*13

               =7*13

               HCF (26, 91) = 13

               LCM (26 , 91) = 2*7*13

                                       =182

NOW,

               LCM*HCF=182*13

                                 =2366

AND,

        THE PRODUCT OF THE TWO NUMBERS= 26*91

                                                                                      =23666

HENCE,

               LCM*HCF=PRODUCT OF THE TWO NUMBERS.

                                                                                                                        VERIFIED.

    (ii). 21, 315.

SOL:                                    21=3*7

                                             315=3*35*7

                         HCF (21,315)=3*7=21

                         LCM(21,315)=3*7*3*5

                                                   =315

NOW,

                         LCM*HCF=315*21=6615

AND,

               THE PRODUCT OF THE TWO NUMBERS=26*91=2366.

HENCE,

               LCM*HCF=PRODUCT OF TWO NUMBERS

                                                                                                         VERIFIED .

 

(iii). 77 , 979.

SOL:                                       77=7*11

                                                979=11*89

                           HCF (77 , 979)= 11

                           LCM(77 , 979)=11*7*89

                                                       =6853

NOW,

                              LCM*HCF=6853*11= 75388

AND,

               THE PRODUCT OF THE TWO NUMBERS = 77*979= 75388.

HENCE,

               LCM*HCF=PRODUCT OF TWO NUMBERS.

                                                                                                         VERIFIED.

 

6. GIVEN THAT HCF (306 , 657) =9, FIND LCM OF 306 AND 657.

SOL:      GIVEN,

                              HCF (306 , 657)=9.

               LET,

                              a= 306  AND b= 657

USING,

                              LCM*HCF=PRODUCT OF TWO NUMBERS.

       ⟹ LCM (306 , 657)=306*657 ÷HCF (306 , 657).

                                             =306*657÷9

                                             =201042÷9

                                             =22338.

7. FIND THE HCF AND LCM OF THE FOLLOWING POSITIVE INTEGERS BY APPLYING THE PRIME             FACTORISATION METHOD:

(i). 24 , 36 , 176.

SOL:      24=2*2*2*3=23*3

               36=2*2*3*3=22*32

               176=2*2*2*2*11=24*112

               HCF (24 , 36 , 176) = 2*2=4

               LCM ( 24 , 36 , 176) = 2*2*2*2*3*3*11= 1584.

                                                     

(ii). 84 , 90 ,120.

SOL:      84=2*2*3*7=22*3*7

               90=3*3*2*5=32*2*5

               120=2*2*2*3*5=23*3*5

               HCF (84 , 90 , 120)= 2*3=6

               LCM (84 , 90 , 120)=2*2*2*3*3*5*7=2520

(iii). 112 , 144 , 168.

SOL:      122= 2*2*2*2*7

               144= 2*2*2*2*3*3

               168=2*2*2*3*7

               HCF (122 , 144 , 168) = 8

               LCM (122 , 144 , 168) = 2*2*2*2*3*3*7=1008

              

8.  HCF OF TWO NUMBERS IS 16 AND THEIR PRODUCT IS 3072. FIND THEIR LCM.

SOL:      GIVEN,

                              HCF OF TWO NUMBERS = 16

                                                         PRODUCT= 3072

               USING,

                              LCM*HCF = PRODUCT OF TWO NUMBERS

                                  ⟹ LCM= 3075÷HCF (16)

                                                 = 3075÷16

                                                =192.

9. FIND THE SMALLEST NUMBERS WHICH WHEN DIVIDED BY 35 , 56 , AND 91 LEAVES REMAINDER 7 IN EACH CASE.




SOL:      HERE,

                              35= 5*7

                              56=2*2*2*7= 237

                              91= 7*13

                          LCM= 2*2*2*5*7*13 = 3640

                          LCM= 3640+7=3647

 

10. FIND THE SMALLEST NUMBER WHICH WHEN INCREASED BY 11 IS EXACTLY DIVISIBLE BY 15 , 20 , 54.

SOL:      LET,

                              THE NUMBERS BE ‘x’

               A/C   TO   QUESTION,

                              x+11 IS DIVISIBLE BY 15 , 20 AND 54.

                              x+11 = LCM(15 , 20 , 54)

NOW,

                              15=3*5

                              20=2*2*5

                              54=2* 3*3*3

                          LCM= 5*3*3*3*2*2

                    ⟹LCM= 540

                 ⟹x+11= 540

                         ⟹x= 540-11                                       (INTERCHANGE)

                                 = 529

 

11.  FIND THE GREATEST NUMBER OF 6 DIGITS EXACTLY DIVISIBLE 24 , 15 AND 36.

SOL:      GREATEST NUMBER OF 6 DIGITS NUMBERS IS 999999

               NOW,

                              24=2*2*2*2*3

                              15=3*5

                              36=2*2*3*3

                         LCM = 3*3*2*2*2*5 = 360.

∴ REQUIRED NUMBER = 999999-REMAINDER WHERE 999999 IS DIVIDED BY 360

                                           = 999999-279

                                           =999720.

 

 

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