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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)
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)
Q. Which of the above are correct?
- a)1 and 2 only
- b)3 and 4 only
- c)1, 2 and 4 only
- d)1, 2, 3 and 4
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Consider the following in respect of natural numbers a, b and c:1. LCM...
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 [∵ p and q are co-prime]
⇒ HCF(ab, ac) = a × HCF (b, c)
⇒ Statement II is correct
LCM (ab, ac) = Product of numbers (ab, ac) /HCF (ab, ac)
⇒ [(atp) × (atq)] / (at)
⇒ atpq
⇒ a [(tp) × (tq)] /t
⇒ a × Product of (b, c) /HCF (b, c)
⇒ a × LCM(b, c)
⇒ Statement I is true
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 [∵ p and q are co-prime]
⇒ HCF(ab, ac) = a × HCF (b, c)
⇒ Statement II is correct
LCM (ab, ac) = Product of numbers (ab, ac) /HCF (ab, ac)
⇒ [(atp) × (atq)] / (at)
⇒ atpq
⇒ a [(tp) × (tq)] /t
⇒ a × Product of (b, c) /HCF (b, c)
⇒ a × LCM(b, c)
⇒ 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
When a = b
⇒HCF (a, a) = a
⇒ LCM (a, a) = a
⇒ LCM (a, a) = HCF (a, a)
⇒ Statement III fails when a = b
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.
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.
Most Upvoted Answer
Consider the following in respect of natural numbers a, b and c:1. LCM...
2. GCD(ab, ac) = a
3. LCM(ab, bc) = ab
4. GCD(ab, bc) = b
5. LCM(ac, bc) = c
6. GCD(ac, bc) = c
Explanation:
1. The least common multiple (LCM) of two numbers is the smallest number that is divisible by both numbers. In this case, LCM(ab, ac) is a because a is a common factor of both ab and ac.
2. The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers. In this case, GCD(ab, ac) is a because a is a common factor of both ab and ac.
3. LCM(ab, bc) is ab because a, b, and c are all prime numbers and do not share any common factors other than 1. Therefore, the LCM is the product of all the numbers involved.
4. GCD(ab, bc) is b because b is a common factor of both ab and bc.
5. LCM(ac, bc) is c because c is a common factor of both ac and bc.
6. GCD(ac, bc) is c because c is a common factor of both ac and bc.
3. LCM(ab, bc) = ab
4. GCD(ab, bc) = b
5. LCM(ac, bc) = c
6. GCD(ac, bc) = c
Explanation:
1. The least common multiple (LCM) of two numbers is the smallest number that is divisible by both numbers. In this case, LCM(ab, ac) is a because a is a common factor of both ab and ac.
2. The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers. In this case, GCD(ab, ac) is a because a is a common factor of both ab and ac.
3. LCM(ab, bc) is ab because a, b, and c are all prime numbers and do not share any common factors other than 1. Therefore, the LCM is the product of all the numbers involved.
4. GCD(ab, bc) is b because b is a common factor of both ab and bc.
5. LCM(ac, bc) is c because c is a common factor of both ac and bc.
6. GCD(ac, bc) is c because c is a common factor of both ac and bc.
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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)Q. Which of the above are correct?a)1 and 2 onlyb)3 and 4 onlyc)1, 2 and 4 onlyd)1, 2, 3 and 4Correct answer is option 'C'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about 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)Q. Which of the above are correct?a)1 and 2 onlyb)3 and 4 onlyc)1, 2 and 4 onlyd)1, 2, 3 and 4Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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)Q. Which of the above are correct?a)1 and 2 onlyb)3 and 4 onlyc)1, 2 and 4 onlyd)1, 2, 3 and 4Correct answer is option 'C'. Can you explain this answer?.
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Solutions for 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)Q. Which of the above are correct?a)1 and 2 onlyb)3 and 4 onlyc)1, 2 and 4 onlyd)1, 2, 3 and 4Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Defence.
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