Vachmi

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1

Express each number as a product of its prime factors:

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

Solution
(i) $140 = 2 × 2 × 5 × 7 = 2^2 × 5 × 7$

(ii) $156 = 2 × 2 × 3 × 13 = 2^2 × 3 × 13$

(iii) $3825 = 3 × 3 × 5 × 5 × 17 = 3^2 × 5^2 × 17$

(iv) $5005 = 5 × 7 × 11 × 13$

(v) $7429 = 17 × 19 × 23$

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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 (ii) 510 and 92 (iii) 336 and 54

Solution
(i) The prime factorisation of 26 and 91 gives :
26 = 2 × 13
91 = 7 × 13

∴ HCF(26, 91) = 13
LCM(26, 91) = 2 × 7 × 13 = 182

Verification:
HCF × LCM = 13 × 182 = 2366
Product of two numbers = 26 × 91 = 2366
Hence verified that LCM × HCF = product of the two numbers

(ii) The prime factorisation of 510 and 92 gives :
510 = 2 × 3 × 5 × 17
92 = 2 × 2 × 23

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

Verification:
HCF × LCM = 2 × 23460 = 46920
Product of two numbers = 510 × 92 = 46920
Hence verified that LCM × HCF = product of the two numbers

(iii) The prime factorisation of 336 and 54 gives :
336 = 2 × 2 × 2 × 2 × 3 × 7
54 = 2 × 3 × 3 × 3

∴ HCF(336, 54) = 2 × 3 = 6
LCM(336, 54) = $2^4$ × $3^3$ × 7 = 3024

Verification:
HCF × LCM = 6 × 3024 = 18144
Product of two numbers = 336 × 54 = 18144
Hence verified that LCM × HCF = product of the two numbers

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3

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

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

Solution
(i) The prime factorisation of 12, 15 and 21 gives :
12 = 2 × 2 × 3
15 = 3 × 5
21 = 3 × 7

∴ HCF(12, 15, 21) = 3
LCM(12, 15, 21) = 2 × 2 × 3 × 5 × 7 = 420

(ii) The prime factorisation of 17, 23 and 29 gives :
17 = 17 × 1
23 = 23 × 1
29 = 29 × 1

∴ HCF(17, 23, 29) = 1
LCM(21, 23, 29) = 17 × 23 × 29 = 11339

(iii) The prime factorisation of 8, 9 and 25 gives :
8 = 2 × 2 × 2
9 = 3 × 3
25 = 5 × 5

∴ HCF(8, 9, 25) = 1
LCM(8, 9, 25) = 8 × 9 × 25 = 1800