LCM and HCF
 The product of the LCM and HCF of two numbers is 24. The difference of the two numbers is 2. Find the numbers ?

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Let the numbers be p and (p + 2).
∴ Product of numbers = HCF × LCM
⇒ p (p + 2) = 24
⇒ p^{2} + 2p – 24 = 0
⇒ p^{2} + 6p – 4p – 24 = 0
⇒ p (p + 6) – 4 (p + 6) = 0
⇒ (p – 4) (p + 6) = 0
⇒ p = 4 or p = – 6Correct Option: D
Let the numbers be p and (p + 2).
∴ Product of numbers = HCF × LCM
⇒ p (p + 2) = 24
⇒ p^{2} + 2p – 24 = 0
⇒ p^{2} + 6p – 4p – 24 = 0
⇒ p (p + 6) – 4 (p + 6) = 0
⇒ (p – 4) (p + 6) = 0
⇒ p = 4 or p = – 6
Numbers = p = 4 and p + 2 = 4 + 2 = 6.
∴ Numbers are 4 and 6.
 The H.C.F. and L.C.M. of two numbers are 21 and 84 respectively. If the ratio the two numbers is 1 : 4, then the larger of the two numbers is

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HCF of numbers = 21
∴ Numbers = 21p and 21q Where p and q are prime to each other.
Ratio of numbers = 1 : 4Correct Option: D
HCF of numbers = 21
∴ Numbers = 21p and 21q Where p and q are prime to each other.
Ratio of numbers = 1 : 4
∴ Larger number = 21 × 4 = 84
 If p : q be the ratio of two whole numbers and r be their HCF, then the LCM of those two numbers is

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We can find the LCM with the help of given formula ,
Product of two numbers = HCF × LCM
⇒ Numbers = rp and rq
∴ rp × rq = r × LCMCorrect Option: D
We can find the LCM with the help of given formula ,
Product of two numbers = HCF × LCM
⇒ Numbers = rp and rq
∴ rp × rq = r × LCM
⇒ LCM = pqr
Hence , the LCM of two numbers is pqr.
 Three numbers are in the ratio 1 : 2 : 3 and their HCF is 12. The numbers are

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Let the numbers are p , 2p and 3p .
Their H.C.F. = p = 12Correct Option: A
Let the numbers are p , 2p and 3p .
Their H.C.F. = p = 12
∴ Numbers = p = 12, 2p = 2 × 12 = 24 and 3p = 3 × 12 = 36
Hence , Three numbers are 12 , 24 and 36 .
 The ratio of two numbers is 3 : 4 and their HCF is 5. Their LCM is :

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If the numbers be 3p and 4p, then
HCF = p = 5
∴ Numbers = 3p = 3 × 5 = 15 and 4p = 4 × 5 = 20Correct Option: B
If the numbers be 3p and 4p, then
HCF = p = 5
∴ Numbers = 3p = 3 × 5 = 15 and 4p = 4 × 5 = 20
∴ LCM = 60