Real Numbers Exercise 1.1 class 10

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Real Numbers Exercise 1.1 class 10

EXERCISE 1.1

1. Express each number as a product of its prime factors:

(i) 140
140 = 2 × 2 × 5 × 7
       = 2² × 5 × 7

(ii) 156
156 = 2 × 2 × 3 × 13
       = 2² × 3 × 13

(iii) 3825
3825 = 3 × 3 × 5 × 5 × 17
         = 3² × 5² × 17

(iv) 5005
5005 = 5 × 7 × 11 × 13

(v) 7429
7429 = 17 × 19 × 23

2. Find LCM and HCF

(i) 26 and 91

26 = 2 × 13
91 = 7 × 13

HCF = 13
LCM = 182

Verification:

LCM × HCF = Product of two numbers

   182 × 13     =   26 × 91 

              2366 = 2366

(ii) 510 and 92

510 = 2 × 3 × 5 × 17
92 = 2 × 2 × 23

HCF = 2

LCM = 2 × 2 × 3 × 5 × 17 × 23

LCM = 23460

Verification:

LCM × HCF = Product of two numbers

23460 × 2      = 510 × 92

         46920   =  46920

(iii) 336 and 54

336 = 2 × 2 × 2 × 2 × 3 × 7

54 = 2 × 3 × 3

HCF = 6

LCM = 2 × 3 × 2 × 2 × 3 × 7

LCM = 3024

Verification:

LCM × HCF = Product of two numbers

3024 ×  6      = 336 × 54

        18144   = 18144

3. Find LCM and HCF

(i) 12, 15, 21

12 = 2 × 2 × 3

15 = 3 × 5

21 = 3 × 7

HCF = 3

LCM = 3 × 2 × 2 × 5 × 7

LCM  = 420

(ii) 17, 23, 29

All are prime numbers.

HCF = 1

LCM = 17 × 23 × 29 

LCM = 11339

(iii) 8, 9, 25

8 = 2 × 2 × 2
9 = 3 × 3
25 = 5 × 5

HCF = 1

LCM = 2 × 2 × 2 × 3 × 3 × 5 × 5

LCM  = 1800

4. Given that HCF (306,657) = 9, find LCM (306,657)

Solution:
LCM × HCF = Product of two numbers
LCM × 9       = 306 × 657
         LCM     = 201042 / 9
         LCM     = 22338

5. Check whether 6n can end with the digit 0 for any natural number n.

Solution:

A number ends with 0 only if it has factor 10 = 2 × 5.

6 = 2 × 3

There is no factor 5.

Therefore 6n can never end with digit 0.

6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

Solution:

(i)

7 × 11 × 13 + 13
= 13(7 × 11 + 1)
= 13 × 78
= 1014

Since it has factors 13 and 78, it is composite.

(ii)

7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
= 5040 + 5
= 5045

5045 = 5 × 1009

So it is composite.

7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field. while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at he starting point?

Solution:

Time taken:

Sonia = 18 min
Ravi = 12 min

They will meet again after LCM of 18 and 12

18 = 2 × 3 × 3
12 = 2 × 2 × 3

LCM = 2 × 3 × 2 × 3

LCM = 36 minutes

Therefore, They meet again after 36 minutes.



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