CAT Exam > CAT Questions > What is the total number of natural numbers ...
Start Learning for Free
What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?
- a)1
- b)2
- c)3
- d)There are no such numbers.
Correct answer is option 'B'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
What is the total number of natural numbers N such that out of all th...
Let S be the smallest factor of N except 1 and L be the largest factor of N except N.
For any natural number N, Smallest factor * largest factor = 2nd smallest factor * 2nd largest factor = 3rd smallest factor x 3rd largest factor = ... = N Thus from the data given in the question and from the above rule, we get, S x L = N, L = 21S. Therefore S x 21S = N. Therefore 21 S2 = N.
Since we want the largest factor (after N) to be 21 times the smallest factor (after 1), we will have to ensure that S takes values less than or equal to the smallest prime factor of 21 i.e. less than or equal to 3.
S = 2, 3
Two such numbers N are possible. And the two values of N are 21 x 22 = 84 and 21 x 32 = 189 Hence, option 2.
Free Test
FREE
| Start Free Test |
Community Answer
What is the total number of natural numbers N such that out of all th...
Understanding the problem
To find the total number of natural numbers N that satisfy the given condition, we need to consider factors of N (excluding 1 and N) where the largest factor is 21 times the smallest factor.
Factors of a number N
- Every number N will have factors in pairs, except when the number is a perfect square.
- For example, factors of 36 are 2, 3, 4, 6, 9, 12, 18. Here, factors are in pairs (2, 18), (3, 12), (4, 9), (6, 6).
- So, the total number of factors of N will be even, unless N is a perfect square.
Applying the condition
- Let the smallest factor be x and the largest factor be 21x.
- We know that x * 21x = 21x^2 = N.
- So, N must be a perfect square for the factors to be in pairs.
- Let N = y^2, where y is a natural number.
Calculating the values
- From N = 21x^2 = y^2, we get x = y/√21.
- For x to be a natural number, y must be a multiple of √21.
- The possible values for y are 1√21, 2√21, 3√21, ...
- Only y = 3√21 and y = 6√21 will give natural numbers for x and N.
Conclusion
- Therefore, the total number of natural numbers N that satisfy the given condition is 2, which corresponds to y = 3√21 and y = 6√21.
- Hence, the correct answer is option B.
To find the total number of natural numbers N that satisfy the given condition, we need to consider factors of N (excluding 1 and N) where the largest factor is 21 times the smallest factor.
Factors of a number N
- Every number N will have factors in pairs, except when the number is a perfect square.
- For example, factors of 36 are 2, 3, 4, 6, 9, 12, 18. Here, factors are in pairs (2, 18), (3, 12), (4, 9), (6, 6).
- So, the total number of factors of N will be even, unless N is a perfect square.
Applying the condition
- Let the smallest factor be x and the largest factor be 21x.
- We know that x * 21x = 21x^2 = N.
- So, N must be a perfect square for the factors to be in pairs.
- Let N = y^2, where y is a natural number.
Calculating the values
- From N = 21x^2 = y^2, we get x = y/√21.
- For x to be a natural number, y must be a multiple of √21.
- The possible values for y are 1√21, 2√21, 3√21, ...
- Only y = 3√21 and y = 6√21 will give natural numbers for x and N.
Conclusion
- Therefore, the total number of natural numbers N that satisfy the given condition is 2, which corresponds to y = 3√21 and y = 6√21.
- Hence, the correct answer is option B.
Attention CAT Students!
To make sure you are not studying endlessly, EduRev has designed CAT study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in CAT.
|
Explore Courses for CAT exam
|
|
Similar CAT Doubts
Top Courses for CATView all
What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.Correct answer is option 'B'. Can you explain this answer?
Question Description
What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.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 What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.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 What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.Correct answer is option 'B'. Can you explain this answer?.
What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.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 What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.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 What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.Correct answer is option 'B'. Can you explain this answer?.
Solutions for What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.Correct answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.Correct answer is option 'B'. Can you explain this answer?, a detailed solution for What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice What is the total number of natural numbers N such that out of all the factors of N (except 1 and N) the largest factor is 21 times the smallest one?a)1b)2c)3d)There are no such numbers.Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice CAT tests.
|
|
Explore Courses for CAT exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup