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Consider the following in respect of natural numbers a, b and c: 1. LCM(ab, ac) = a × LCM(b, c) 2. HCF(ab, ac) = a × HCF(b, c) 3. HCF (a, b) < LCM(a, b) 4. HCF(a, b) divides LCM(a, b) Where, a ≠ b ≠ c  Which of the above are correct?

Consider the following in respect of natural numbers a, b and c: 1. LCM(ab, ac) = a × LCM(b, c) 2. HCF(ab, ac) = a × HCF(b, c) 3. HCF (a, b) < LCM(a, b) 4. HCF(a, b) divides LCM(a, b) Where, a ≠ b ≠ c  Which of the above are correct? Correct Answer 1, 2, 3 and 4

Statement II:

Let the HCF of b and c be t

⇒ b = tp and c = tq where p and q are co-prime numbers

⇒ ab = atp and ac = atq

⇒ HCF(ab, ac) = at                 

⇒ HCF(ab, ac) = a × HCF (b, c)

⇒ Statement II is correct

LCM (ab, ac) = Product of numbers (ab, ac) /HCF (ab, ac)

⇒ / (at)

⇒ atpq

⇒ a /t

⇒ a × Product of (b, c) /HCF (b, c)

⇒ a × LCM(b, c)

So, Statement I is true

Statement III:

When a = b

⇒HCF (a, a) = a

⇒ LCM (a, a) = a

⇒ LCM (a, a) = HCF (a, a)

⇒ Statement III fails when a = b

But according to question a ≠ b ≠ c 

So, Statement III is also correct.

Statement IV:

Let the HCF (a, b) = r

⇒ a = rp and b = rq, where r and q are co prime

LCM = Product of the numbers/HCF

⇒ rp × rq/r = rpq

⇒ LCM = HCF × pq

⇒ HCF divides LCM

⇒ Statement IV is correct

∴ Statement 1, 2 and 4 are true.

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