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We have two unknown positive integers m and n, whose product is less than 100.
There are two additional statements of facts available:
mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }
m + n is a perfect square.
Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?
- a)Each statement alone is sufficient to answer the question.
- b)Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.
- c)Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.
- d)Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.
- e)Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.
Correct answer is option 'B'. Can you explain this answer?
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We have two unknown positive integers m and n, whose product is less ...
Given, m and n are two positive integers having product less than 100
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Statement 1:
mn is divisible by 6 consecutive integers. This means mn must be divisible by the LCM of these 6 integers multiples.
Only numbers from 1 to 6 satisfy this with LCM 60. Any other set of 6 consecutive integers clearly exceeds 100 as its LCM.
Only such a number satisfying the condition is 60.
60 can further be expressed as the product of two positive integers in the following ways:
1 x 60, 2 x 30, 3 x 20, 4 x 15, 5 x 12 and 6 x 10.
So, no unique values of m and n can be determined from the above.
Statement 2:
m + n is a perfect square.
From this statement alone too no unique set of solutions can be determined.
From Statements 1 and 2: Only the pair 6 x 10 satisfies the condition.
Hence, both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.
Most Upvoted Answer
We have two unknown positive integers m and n, whose product is less ...
Given:
We have two unknown positive integers m and n, whose product is less than 100.
mn is divisible by six consecutive integers { j, j+1,...,j+5 }
m*n is a perfect square.
To find:
Which of the two statements above, alone or in combination, is sufficient to determine the numbers m and n?
Solution:
Let's analyze each statement separately and then combine them to see if we can determine the values of m and n.
Statement 1: mn is divisible by six consecutive integers { j, j+1,...,j+5 }
This statement implies that mn is divisible by 6! (factorial of 6) because it is divisible by six consecutive integers. In other words, mn is divisible by 6*5*4*3*2*1 = 720.
There are two possibilities for mn:
1. mn = 720k, where k is a positive integer.
2. mn = 720k', where k' is a positive fraction less than 1.
Now, let's consider the values of m and n separately:
Case 1:
If mn = 720k, then both m and n can be factors of 720k. Since the product of m and n is less than 100, we need to find two factors of 720k whose product is less than 100.
The prime factorization of 720 is: 2^4 * 3^2 * 5.
Possible pairs of factors of 720 are:
- (1, 720)
- (2, 360)
- (3, 240)
- (4, 180)
- (5, 144)
- (6, 120)
- (8, 90)
- (9, 80)
- (10, 72)
- (12, 60)
- (15, 48)
- (16, 45)
- (18, 40)
- (20, 36)
- (24, 30)
Out of these pairs, we need to find a pair whose product is a perfect square. Since we don't have any information about the values of m and n, we cannot determine which pair satisfies this condition. Therefore, statement 1 alone is not sufficient to determine the values of m and n.
Case 2:
If mn = 720k', where k' is a positive fraction less than 1, then both m and n are fractions. Since the question specifies that m and n are positive integers, this case is not possible. Therefore, we can ignore this case.
Statement 2: m*n is a perfect square
This statement implies that the product of m and n is a perfect square. Let's analyze this statement further.
If m*n is a perfect square, then both m and n must be perfect squares themselves. We can represent m and n as m = p^2 and n = q^2, where p and q are positive integers.
Now, we can rewrite the product mn as mn = p^2 * q^2 = (pq)^2, which is a perfect square.
Since both m and n are perfect squares, we can find their values by finding the square roots of mn. Therefore, statement 2 alone is
We have two unknown positive integers m and n, whose product is less than 100.
mn is divisible by six consecutive integers { j, j+1,...,j+5 }
m*n is a perfect square.
To find:
Which of the two statements above, alone or in combination, is sufficient to determine the numbers m and n?
Solution:
Let's analyze each statement separately and then combine them to see if we can determine the values of m and n.
Statement 1: mn is divisible by six consecutive integers { j, j+1,...,j+5 }
This statement implies that mn is divisible by 6! (factorial of 6) because it is divisible by six consecutive integers. In other words, mn is divisible by 6*5*4*3*2*1 = 720.
There are two possibilities for mn:
1. mn = 720k, where k is a positive integer.
2. mn = 720k', where k' is a positive fraction less than 1.
Now, let's consider the values of m and n separately:
Case 1:
If mn = 720k, then both m and n can be factors of 720k. Since the product of m and n is less than 100, we need to find two factors of 720k whose product is less than 100.
The prime factorization of 720 is: 2^4 * 3^2 * 5.
Possible pairs of factors of 720 are:
- (1, 720)
- (2, 360)
- (3, 240)
- (4, 180)
- (5, 144)
- (6, 120)
- (8, 90)
- (9, 80)
- (10, 72)
- (12, 60)
- (15, 48)
- (16, 45)
- (18, 40)
- (20, 36)
- (24, 30)
Out of these pairs, we need to find a pair whose product is a perfect square. Since we don't have any information about the values of m and n, we cannot determine which pair satisfies this condition. Therefore, statement 1 alone is not sufficient to determine the values of m and n.
Case 2:
If mn = 720k', where k' is a positive fraction less than 1, then both m and n are fractions. Since the question specifies that m and n are positive integers, this case is not possible. Therefore, we can ignore this case.
Statement 2: m*n is a perfect square
This statement implies that the product of m and n is a perfect square. Let's analyze this statement further.
If m*n is a perfect square, then both m and n must be perfect squares themselves. We can represent m and n as m = p^2 and n = q^2, where p and q are positive integers.
Now, we can rewrite the product mn as mn = p^2 * q^2 = (pq)^2, which is a perfect square.
Since both m and n are perfect squares, we can find their values by finding the square roots of mn. Therefore, statement 2 alone is
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We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer?
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We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer?.
We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer?.
Solutions for We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
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We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer?, a detailed solution for We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice We have two unknown positive integers m and n, whose product is less than 100.There are two additional statements of facts available:mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }m + n is a perfect square.Q. Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?a) Each statement alone is sufficient to answer the question.b) Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.c) Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.d) Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.e) Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice CAT tests.
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