GMAT Exam > GMAT Questions > If x and y are consecutive positive integer m...
Start Learning for Free
If x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?
- a)27
- b)18
- c)9
- d)6
- e)3
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If x and y are consecutive positive integer multiples of 3, what is th...
Understanding the Problem
We need to find the greatest integer j such that the expression xy/j is always an integer, where x and y are consecutive positive integer multiples of 3.
Defining x and y
- Let x = 3n (first multiple of 3)
- Let y = 3(n + 1) (next consecutive multiple of 3)
Calculating xy
- The product xy can be expressed as:
xy = (3n)(3(n + 1)) = 9n(n + 1)
Factors of xy
- The expression 9n(n + 1) consists of:
- 9 (which is 3^2)
- n (a positive integer)
- (n + 1) (the next consecutive integer, hence coprime to n)
Finding j
- To ensure xy/j is always an integer, j must divide 9n(n + 1).
- The factors of 9 are 1, 3, and 9.
Analyzing Divisibility
- Since n and n + 1 are consecutive integers, at least one of them is always even.
- Therefore, the product n(n + 1) is even, ensuring xy is always divisible by 2.
Determining the Greatest j
- The product xy is divisible by 9 (from the factor 9).
- The minimum j that can provide an integer result for xy/j is the highest factor of 9 that is still guaranteed to divide the product.
Conclusion
- The greatest integer j such that xy/j is always an integer is 9. However, we need to ensure it divides 9n(n + 1) for all positive integers n, which leads us to consider the factors of 18.
Thus, the correct answer is option B (18), as it encompasses all scenarios where n(n + 1) remains even.
We need to find the greatest integer j such that the expression xy/j is always an integer, where x and y are consecutive positive integer multiples of 3.
Defining x and y
- Let x = 3n (first multiple of 3)
- Let y = 3(n + 1) (next consecutive multiple of 3)
Calculating xy
- The product xy can be expressed as:
xy = (3n)(3(n + 1)) = 9n(n + 1)
Factors of xy
- The expression 9n(n + 1) consists of:
- 9 (which is 3^2)
- n (a positive integer)
- (n + 1) (the next consecutive integer, hence coprime to n)
Finding j
- To ensure xy/j is always an integer, j must divide 9n(n + 1).
- The factors of 9 are 1, 3, and 9.
Analyzing Divisibility
- Since n and n + 1 are consecutive integers, at least one of them is always even.
- Therefore, the product n(n + 1) is even, ensuring xy is always divisible by 2.
Determining the Greatest j
- The product xy is divisible by 9 (from the factor 9).
- The minimum j that can provide an integer result for xy/j is the highest factor of 9 that is still guaranteed to divide the product.
Conclusion
- The greatest integer j such that xy/j is always an integer is 9. However, we need to ensure it divides 9n(n + 1) for all positive integers n, which leads us to consider the factors of 18.
Thus, the correct answer is option B (18), as it encompasses all scenarios where n(n + 1) remains even.
Free Test
FREE
| Start Free Test |
Community Answer
If x and y are consecutive positive integer multiples of 3, what is th...
Understanding the Problem
To solve the problem, we need to identify consecutive positive integer multiples of 3, denoted as x and y.
Identifying x and y
- Let x = 3n (the first multiple of 3)
- Consequently, y = 3(n + 1) (the next consecutive multiple of 3)
Thus, we can express the product xy as follows:
Calculating the Product xy
- xy = (3n) * (3(n + 1)) = 9n(n + 1)
Finding the Divisor j
We want to find the greatest integer j such that xy/j is an integer for any positive integer n. To do this, we will analyze the factors of 9n(n + 1).
- The term n(n + 1) consists of two consecutive integers, ensuring that one of them is even. Hence, n(n + 1) is always at least divisible by 2.
Determining the Greatest j
- The product xy = 9n(n + 1) contains:
- A factor of 9 (which is 3^2)
- A factor of at least 2 from n(n + 1)
Therefore, the total product can be expressed in terms of its prime factors:
- xy = 2^1 * 3^2 * k (where k is some integer based on n)
To find the greatest j that divides xy, we need the maximum constant divisor that can be divided out:
- The constant factors are 2 and 9.
Combining these factors, the highest j that divides xy for any n is:
Conclusion
- The greatest integer j is 18 (2 * 9), which means j = 18 fits all conditions required to ensure that xy/j is always an integer.
Thus, the answer is option 'B': 18.
To solve the problem, we need to identify consecutive positive integer multiples of 3, denoted as x and y.
Identifying x and y
- Let x = 3n (the first multiple of 3)
- Consequently, y = 3(n + 1) (the next consecutive multiple of 3)
Thus, we can express the product xy as follows:
Calculating the Product xy
- xy = (3n) * (3(n + 1)) = 9n(n + 1)
Finding the Divisor j
We want to find the greatest integer j such that xy/j is an integer for any positive integer n. To do this, we will analyze the factors of 9n(n + 1).
- The term n(n + 1) consists of two consecutive integers, ensuring that one of them is even. Hence, n(n + 1) is always at least divisible by 2.
Determining the Greatest j
- The product xy = 9n(n + 1) contains:
- A factor of 9 (which is 3^2)
- A factor of at least 2 from n(n + 1)
Therefore, the total product can be expressed in terms of its prime factors:
- xy = 2^1 * 3^2 * k (where k is some integer based on n)
To find the greatest j that divides xy, we need the maximum constant divisor that can be divided out:
- The constant factors are 2 and 9.
Combining these factors, the highest j that divides xy for any n is:
Conclusion
- The greatest integer j is 18 (2 * 9), which means j = 18 fits all conditions required to ensure that xy/j is always an integer.
Thus, the answer is option 'B': 18.
|
Explore Courses for GMAT exam
|
|
Top Courses for GMATView all
Question Description
If x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. 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 x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. 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 x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. Can you explain this answer?.
If x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. 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 x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. 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 x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. Can you explain this answer?.
Solutions for If x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for GMAT.
Download more important topics, notes, lectures and mock test series for GMAT Exam by signing up for free.
Here you can find the meaning of If x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. Can you explain this answer?, a detailed solution for If x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of If x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If x and y are consecutive positive integer multiples of 3, what is the greatest integer j such that xy/j is always an integer?a)27b)18c)9d)6e)3Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice GMAT tests.
|
|
Explore Courses for GMAT exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up
within 7 days!
within 7 days!
Takes less than 10 seconds to signup