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If the LCM of two numbers a and b is 1104 and their HCF is 4, which of the following MUST be true?
I. a * b = 4416
II. a and b are both divisible by 8
III. a : b = 48 : 23 or a : b = 23 : 48
I. a * b = 4416
II. a and b are both divisible by 8
III. a : b = 48 : 23 or a : b = 23 : 48
- a)I and III only
- b)II only
- c)I only
- d)II and III only
- e)I, II, and III
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If the LCM of two numbers a and b is 1104 and their HCF is 4, which of...
Before we start solving the question in hand, here are a couple of important concepts that we need to know.
Concept 1:
Product of two numbers is the same as the product of the LCM and HCF of those two numbers.
i.e., If the numbers are a and b, a * b = LCM (a, b) * HCF (a, b)
Note: a * b * c NEED NOT be equal to LCM(a, b, c) * HCF(a, b, c).
This rule works for 2 numbers, irrespective of whether the numbers are both integers, both fractions, one fraction and the other an integer.
Concept 2:
Let ‘h’ be the HCF of a and b and ‘L’ be the LCM of a and b.
a can be expressed as m*h and b can be expressed as n*h because h is a factor common to both the numbers.
a = mh and b = nh.
a can be expressed as m*h and b can be expressed as n*h because h is a factor common to both the numbers.
a = mh and b = nh.
Note, m and n are co-prime (have no factor in common) because ‘h’ is the HCF of the two numbers. HCF of two numbers holds all factors common to both the numbers.Hence, we can deduce that the LCM (a, b), L = m*n*hi.e., the HCF of two numbers will be a factor of the LCM of the two numbers.
Data given in the question stem:
LCM of a and b is 1104 and their HCF is 4.
Statement I: a * b = 4416
Result 1 states that a * b = LCM (a, b) * HCF (a, b).
So, a * b = 1104 * 4 = 4416.
Statement I is true.
Statement II: a and b are both divisible by 8
The HCF of a and b is 4. So, the largest number that could divide both a and b is 4.
If 8 could divide both a and b, the largest number that could divide both would have been 8.
Consequently, the HCF of the two numbers would have been 8 and not 4.
So, statement II is NOT true.
Statement III: a : b = 48 : 23 or a : b = 23 : 48
Result 2 comes in handy to evaluate statement III.
If L is the LCM(a, b) and h is the HCF(a, b), L = m * n * h.
Where a = mh and b = nh and m and n are co-prime.
We have to determine whether a : b = 48 : 23 or 23 : 48.
i.e., we have to determine whether m : n = 48 : 23 or 23 : 48.
Because L = m * n * h, 1104 = m * n * 4
Or m * n = 1104/4 = 276
Note: m and n are co-prime.
If m and n are 48 and 23 or vice versa, m * n = 1104 and not 276.
Statement III is NOT true.
Most Upvoted Answer
If the LCM of two numbers a and b is 1104 and their HCF is 4, which of...
Given: The product of x, y and HCF(x,y) = 1080
To find: Number of pairs of integers (x,y)
Approach:
- Find the prime factors of 1080
- Use the prime factors to form pairs of x and y such that their HCF is a factor of 1080
- Count the number of such pairs
Prime factorization of 1080 = 2^3 * 3^3 * 5
- HCF(x,y) can be any one of the factors of 1080
- Let's consider the case where HCF(x,y) = 2^0 (i.e., HCF = 1)
- In this case, x and y can have any of the prime factors of 1080 except 2
- There are 2 possibilities for each prime factor (either it can be a factor of x or y)
- So, the number of pairs of x and y in this case = 2^3 * 3^3 * 5^2 = 540
- Similarly, we can consider the cases where HCF(x,y) = 2, 2^2, 2^3, 3, 3^2, 5, 2*3, 2*5, 3*5
- For each case, we need to find the number of pairs of x and y such that their HCF is the given factor
- For example, when HCF(x,y) = 2, x and y can have any of the prime factors of 1080 except 2^2 and 2^3
- There are 2 possibilities for each prime factor (either it can be a factor of x or y)
- So, the number of pairs of x and y in this case = 2^2 * 3^3 * 5^2 = 540
- Similarly, we can find the number of pairs of x and y for each case
- Finally, we add up the number of pairs from all cases to get the total number of pairs of x and y
- Total number of pairs of x and y = 540 + 135 + 45 + 27 + 90 + 30 + 54 + 18 + 36 + 18 = 963
- But we have only counted the cases where x and y are positive integers
- To account for negative integers, we can simply double the count (since the product of two negative integers is positive)
- So, the actual number of pairs of x and y = 2 * 963 = 1926
- But we have counted some cases twice (where x and y are both positive and negative)
- These cases occur when HCF(x,y) = 2, 2*3, 2*5, 3*5
- So, we need to subtract the number of such cases
- Number of such cases = 135 + 90 + 36 + 18 = 279
- Actual number of pairs of x and y = 1926 - 279 = 1647
Answer: Option (C) 9
To find: Number of pairs of integers (x,y)
Approach:
- Find the prime factors of 1080
- Use the prime factors to form pairs of x and y such that their HCF is a factor of 1080
- Count the number of such pairs
Prime factorization of 1080 = 2^3 * 3^3 * 5
- HCF(x,y) can be any one of the factors of 1080
- Let's consider the case where HCF(x,y) = 2^0 (i.e., HCF = 1)
- In this case, x and y can have any of the prime factors of 1080 except 2
- There are 2 possibilities for each prime factor (either it can be a factor of x or y)
- So, the number of pairs of x and y in this case = 2^3 * 3^3 * 5^2 = 540
- Similarly, we can consider the cases where HCF(x,y) = 2, 2^2, 2^3, 3, 3^2, 5, 2*3, 2*5, 3*5
- For each case, we need to find the number of pairs of x and y such that their HCF is the given factor
- For example, when HCF(x,y) = 2, x and y can have any of the prime factors of 1080 except 2^2 and 2^3
- There are 2 possibilities for each prime factor (either it can be a factor of x or y)
- So, the number of pairs of x and y in this case = 2^2 * 3^3 * 5^2 = 540
- Similarly, we can find the number of pairs of x and y for each case
- Finally, we add up the number of pairs from all cases to get the total number of pairs of x and y
- Total number of pairs of x and y = 540 + 135 + 45 + 27 + 90 + 30 + 54 + 18 + 36 + 18 = 963
- But we have only counted the cases where x and y are positive integers
- To account for negative integers, we can simply double the count (since the product of two negative integers is positive)
- So, the actual number of pairs of x and y = 2 * 963 = 1926
- But we have counted some cases twice (where x and y are both positive and negative)
- These cases occur when HCF(x,y) = 2, 2*3, 2*5, 3*5
- So, we need to subtract the number of such cases
- Number of such cases = 135 + 90 + 36 + 18 = 279
- Actual number of pairs of x and y = 1926 - 279 = 1647
Answer: Option (C) 9
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If the LCM of two numbers a and b is 1104 and their HCF is 4, which of the following MUST be true?I. a * b = 4416II. a and b are both divisible by 8III. a : b = 48 : 23 or a : b = 23 : 48a)I and III onlyb)II onlyc)I onlyd)II and III onlye)I, II, and IIICorrect answer is option 'C'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If the LCM of two numbers a and b is 1104 and their HCF is 4, which of the following MUST be true?I. a * b = 4416II. a and b are both divisible by 8III. a : b = 48 : 23 or a : b = 23 : 48a)I and III onlyb)II onlyc)I onlyd)II and III onlye)I, II, and IIICorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the LCM of two numbers a and b is 1104 and their HCF is 4, which of the following MUST be true?I. a * b = 4416II. a and b are both divisible by 8III. a : b = 48 : 23 or a : b = 23 : 48a)I and III onlyb)II onlyc)I onlyd)II and III onlye)I, II, and IIICorrect answer is option 'C'. Can you explain this answer?.
If the LCM of two numbers a and b is 1104 and their HCF is 4, which of the following MUST be true?I. a * b = 4416II. a and b are both divisible by 8III. a : b = 48 : 23 or a : b = 23 : 48a)I and III onlyb)II onlyc)I onlyd)II and III onlye)I, II, and IIICorrect answer is option 'C'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If the LCM of two numbers a and b is 1104 and their HCF is 4, which of the following MUST be true?I. a * b = 4416II. a and b are both divisible by 8III. a : b = 48 : 23 or a : b = 23 : 48a)I and III onlyb)II onlyc)I onlyd)II and III onlye)I, II, and IIICorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the LCM of two numbers a and b is 1104 and their HCF is 4, which of the following MUST be true?I. a * b = 4416II. a and b are both divisible by 8III. a : b = 48 : 23 or a : b = 23 : 48a)I and III onlyb)II onlyc)I onlyd)II and III onlye)I, II, and IIICorrect answer is option 'C'. Can you explain this answer?.
Solutions for If the LCM of two numbers a and b is 1104 and their HCF is 4, which of the following MUST be true?I. a * b = 4416II. a and b are both divisible by 8III. a : b = 48 : 23 or a : b = 23 : 48a)I and III onlyb)II onlyc)I onlyd)II and III onlye)I, II, and IIICorrect answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for GMAT.
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If the LCM of two numbers a and b is 1104 and their HCF is 4, which of the following MUST be true?I. a * b = 4416II. a and b are both divisible by 8III. a : b = 48 : 23 or a : b = 23 : 48a)I and III onlyb)II onlyc)I onlyd)II and III onlye)I, II, and IIICorrect answer is option 'C'. Can you explain this answer?, a detailed solution for If the LCM of two numbers a and b is 1104 and their HCF is 4, which of the following MUST be true?I. a * b = 4416II. a and b are both divisible by 8III. a : b = 48 : 23 or a : b = 23 : 48a)I and III onlyb)II onlyc)I onlyd)II and III onlye)I, II, and IIICorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of If the LCM of two numbers a and b is 1104 and their HCF is 4, which of the following MUST be true?I. a * b = 4416II. a and b are both divisible by 8III. a : b = 48 : 23 or a : b = 23 : 48a)I and III onlyb)II onlyc)I onlyd)II and III onlye)I, II, and IIICorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
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