**Q7 ** 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?

The time after which they meet again at the starting point will be equal to the LCM of the times they individually take to complete one round.
Time taken by Sonia = 18 = 2 x 32
Time taken by Ravi = 12 = 22 x 3
LCM(18,12) = 22 x 32 = 36
Therefore they would again meet at the starting point after 36 minutes.

**Q6** Explain why 7 11 13 + 13 and 7 6 5 4 3 2 1 + 5 are composite numbers.

7 x 11 x 13 + 13
= (7 x 11 + 1) x 13
= 78 x 13
= 2 x 3 x 132
7 x 6 x 5 x 4 x 3 x 2 x 1 + 5
= (7 x 6 x 4 x 3 x 2 x 1 + 1) x 5
= 5 x 1008
After Solving we observed that both the number are even numbers and the number rule says that we can take atleast two common out of two numbers. So that the number is composite number.

**Q5** Check whether can end with the digit 0 for any natural number n.

By prime factorizing we have
6n = 2n x 3n
A number will end with 0 if it has at least 1 as the power of both 2 and 5 in its prime factorization. Since the power of 5 is 0 in the prime factorization of 6n we can conclude that for no value of n 6n will end with the digit 0.

**Q4 **Given that HCF (306, 657) = 9, find LCM (306, 657).

As we know the product of HCF and LCM of two numbers is equal to the product of the two numbers we have
HCF (306, 657) x LCM (306, 657) = 306 x 657

**Q3 (3) ** Find the LCM and HCF of the following integers by applying the prime factorisation method. 8, 9 and 25

The given numbers are written as the product of their prime factors as follows
8 = 23
9 = 32
25 = 52
HCF = 1
LCM = 23 x 32 x 52 = 1800

**Q3 (2) ** Find the LCM and HCF of the following integers by applying the prime factorisation method. 17, 23 and 29

The given numbers are written as the product of their prime factors as follows
17 = 1 x 17
23 = 1 x 23
29 = 1 x 29
HCF = 1
LCM = 17 x 23 x 29 = 11339

**Q3 (1) ** Find the LCM and HCF of the following integers by applying the prime factorisation method. 12, 15 and 21

The numbers can be written as the product of their prime factors as follows
12 = 22 x 3
15 = 3 x 5
21 = 3 x 7
HCF = 3
LCM = 22 x 3 x 5 x 7 = 420

**Q2 (3) ** Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. 336 and 54

336 is expressed as the product of its prime factor as
336 = 24 x 3 x 7
54 is expressed as the product of its prime factor as
54 = 2 x 33
HCF(336,54) = 2 x 3 = 6
LCM(336,54) = 24 x 33 x 7 = 3024
HCF x LCM = 6 x 3024 = 18144
336 x 54 = 18144
336 x 54 = HCF x LCM
Hence Verified

**Q2 (2) ** Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. 510 and 92

The number can be expressed as the product of prime factors as
510 = 2 x 3 x 5 x 17
92 = 22 x 23
HCF(510,92) = 2
LCM(510,92) = 22 x 3 x 5 x 17 x 23 = 23460
HCF x LCM = 2 x 23460 = 46920
510 x 92 = 46920
510 x 92 = HCF x LCM
Hence Verified

**Q2 (1) ** Find the LCM and HCF of the following pairs of integers and verify that LCM HCF =

product of the two numbers: 26 and 91

26 = 2 x 13
91 = 7 x 13
HCF(26,91) = 13
LCM(26,91) = 2 x 7 x 13 = 182
HCF x LCM = 13 x 182 = 2366
26 x 91 = 2366
26 x 91 = HCF x LCM
Hence Verified

**Q1 (5) ** Express each number as a product of its prime factors: 7429

The given number can be expressed as the product of their prime factors as follows

** Q1 (4) ** Express each number as a product of its prime factors: 5005

The given number can be expressed as the product of its prime factors as follows.

** Q1 (3) ** Express each number as a product of its prime factors: 3825

The number is expressed as the product of the prime factors as follows

**Q 1 (2)** Express each number as a product of its prime factors: 156

The given number can be expressed as follows

**Q1 (1) ** Express each number as a product of its prime factors: 140

The number can be as a product of its prime factors as follows

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