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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 $\times$ 11 $\times$ 13 + 13 and 7 $\times$  6 $\times$  5 $\times$ 4 $\times$ 3 $\times$ 2 $\times$ 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 $6 ^n$  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 $\times$ 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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