# Exercise 1.2

Question 1

Express each number as product of its prime factors:

(i) 140

Sol :  (ii) 156

Sol :  (iii) 3825

Sol :  (iv) 5005

Sol : (v) 7429

Sol : Question 2

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

(i) 26 and 91

Sol :  HCF = 13

LCM = Product of the two numbers  = 2366

Hence,Product of the two numbers = HCF × LCM

(ii) 510 and 92

Sol :  HCF = 2

LCM = Product of the two numbers  = 46920

Hence,Product of the two numbers = HCF × LCM

(iii) 336 and 54

Sol :      = 3024

Product of the two numbers  = 18144

Hence,Product of the two numbers = HCF × LCM

Question 3

Find the LCM and HCF of the following integers by applying the prime factorization method.

(i) 12 , 15 and 21

Sol :   HCF = 3

LCM (ii) 17 , 23 and 29

Sol :   HCF = 1

LCM (iii) 8 , 9 and 25

Sol :   HCF = 1

LCM = 11339

Question 4

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

Sol :

We know that , LCM × HCF = Product of the two numbers  LCM  LCM = 22338

Question 5

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

Sol :

If any number ends with the digit 0, it should be divisible by 10 or in other words, it will also be divisible by 2 and 5 as Prime factorisation of It can be observed that 5 is not in the prime factorisation of .

Hence, for any value of n , will not be divisible by 5 .

Therefore, cannot end with the digit 0 for any natural number n .

Question 6

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

Sol :

Numbers are of two types – prime and composite. Prime numbers can be divided by 1 and only itself, whereas composite numbers have factors other than 1 and itself.

It can be observed that     The given expression has 6 and 13 as its factors. Therefore, it is a composite number.    1009 cannot be factorised further. therefore, the given expression has 5 and 1009 as its factors.  Hence ,it is a composite number .

Question 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 the starting point?

Sol :

It can be observed that Ravi takes lesser time than Sonia for completing 1 round of the circular path. As they are going in the same direction, they will meet again at the same time when Ravi will have completed 1 round of that circular path with respect to Sonia. And the total time taken for completing this 1 round of circular path will be the LCM of time taken by Sonia and Ravi for completing 1 round of circular path respectively i.e., LCM of 18 minutes and 12 minutes.  LCM of 12 and 18 Therefore, Ravi and Sonia will meet together at the starting point after 36 minutes.

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