(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

(i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

26 = 2 × 13 and 91 = 7 × 13

LCM (26, 91) = 2 × 7 × 13 = 182**HCF (26, 91) = 13 Now, LCM × HCF = 182 × 13 = 2366 and 26 × 91 = 2366****i.e., LCM × HCF = Product of two numbers.**

(ii) The prime factorisation of 510 and 92 is,

510 = 2 × 3 × 5 × 17 and 92 = 2 × 2 × 23

LCM (510, 92) = 2 × 2 × 3 × 5 × 17 × 23 = 23460

HCF (510, 92) = 2

Now, LCM × HCF = 23460 × 2 = 46920

and 510 × 92 = 46920

i.e., LCM × HCF = Product of two numbers.

(iii) The prime factorisation of 336 and 54 is,

336 = 2 × 2 × 2 × 2 × 3 × 7 and 54 = 2 × 3 × 3 × 3

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

and HCF(336, 54) = 2 × 3 = 6

Now, LCM × HCF = 3024 × 6 = 18144

Also, 336 × 54 = 18144

Thus, LCM × HCF = Product of two numbers.

LCM × 9 = 306× 657

⇒ LCM =306 × 657/ 9

= 22338

Thus, LCM of 306 and 657 is 22338

Here, n is a natural number and let 6^{n} end with digit 0.

6^{n} is divisible by 5.

But the prime factors of 6 are 2 and 3. i.e., 6 = 2 × 3

⇒ 6^{n} = (2 × 3)^{n}

i.e., In the prime factorization of 6^{n}, there is no factor 5.

So, by the fundamental theorem of Arithmetic, every composite number can be expressed as a product of primes and this factorisation is unique apart from the order in which the prime factorization occurs.

Our assumption that 6^{n} ends with digit 0, is wrong.

Thus, there does not exist any natural number n for which 6^{n} ends with zero.

We have

7 × 11 × 13 + 13 = 13((7 × 11) + 1) = 13(78), which cannot

be a prime number because it has factors 13 and 78.

Also, 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5

= 5[7 × 6 × 4 × 3 × 2 × 1 + 1],

which is also not a prime number because it has a factor 5

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

Time taken by Sonia to drive one round of the field = 18 minutes

Time taken by Ravi to drive one round of the field = 12 minutes

LCM of 18 and 12 gives the exact number of minutes after which they meet again at the starting point.

Now, 18 = 2 × 3 × 3 and 12 = 2 × 2 × 3

LCM of 18 and 12 = 2 × 2 × 3 × 3 = 36

Thus, they will meet again at the starting point after 36 minutes

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