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DIRECTIONS for the question: Solve the following question and mark the best possible option.
If the expression x2 + 9 and x + 5 are to have a common odd factor other than 1 such that x is a natural number less than 301, then how many values of x are possible.
- a)16
- b)17
- c)18
- d)15
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
DIRECTIONS for the question:Solve the following question and mark the ...
Let the required common odd factor be K, now x2 + 9 = (x + 5)2 – (10x + 16).
Now 10x + 16 must also be divisible by K, 10x + 16 = 10(x+5) – 34. So 34 must also be divisible by K. So k clearly has to be 17.
Now x + 5 is divisible by 17, so x + 5 = 17, 34, 51, 68 and so on, which gives x = 12, 29, 46, 63, 80, 97 ...284.
So there will be total 17 values.
So there will be total 17 values.
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Community Answer
DIRECTIONS for the question:Solve the following question and mark the ...
Solution:
To find the possible values of x, we need to find the common odd factors of the expressions x^2 - 9 and x - 5.
Factoring the expressions:
1. x^2 - 9 = (x + 3)(x - 3)
2. x - 5
Finding the common factors:
To find the common factors, we need to find the values of x for which (x + 3)(x - 3) and x - 5 have a common odd factor.
Case 1: x + 3 and x - 5 have a common odd factor:
In this case, we need to find the values of x such that (x + 3)(x - 3) is divisible by an odd number.
Since (x + 3)(x - 3) is the product of two consecutive odd numbers, it is always divisible by 2. Therefore, for (x + 3)(x - 3) to be divisible by an odd number, x + 3 and x - 3 must both be divisible by an odd number.
Considering the values of x less than 301, we can see that the only odd number that divides both x + 3 and x - 3 is 1.
Case 2: x - 5 has a common odd factor with x^2 - 9:
In this case, we need to find the values of x such that x - 5 is divisible by an odd number and x^2 - 9 is divisible by the same odd number.
Since x - 5 is the difference of two consecutive odd numbers, it is always divisible by 2. Therefore, for x - 5 to be divisible by an odd number, x must be an odd number.
Now, let's consider the values of x less than 301 that are odd.
Calculating the number of possible values:
From case 1, we found that there are no values of x for which (x + 3)(x - 3) has a common odd factor.
From case 2, we found that there are 17 values of x less than 301 that are odd.
Therefore, the total number of values of x that satisfy the given conditions is 17.
Conclusion:
Hence, the correct answer is option 'B' (17).
To find the possible values of x, we need to find the common odd factors of the expressions x^2 - 9 and x - 5.
Factoring the expressions:
1. x^2 - 9 = (x + 3)(x - 3)
2. x - 5
Finding the common factors:
To find the common factors, we need to find the values of x for which (x + 3)(x - 3) and x - 5 have a common odd factor.
Case 1: x + 3 and x - 5 have a common odd factor:
In this case, we need to find the values of x such that (x + 3)(x - 3) is divisible by an odd number.
Since (x + 3)(x - 3) is the product of two consecutive odd numbers, it is always divisible by 2. Therefore, for (x + 3)(x - 3) to be divisible by an odd number, x + 3 and x - 3 must both be divisible by an odd number.
Considering the values of x less than 301, we can see that the only odd number that divides both x + 3 and x - 3 is 1.
Case 2: x - 5 has a common odd factor with x^2 - 9:
In this case, we need to find the values of x such that x - 5 is divisible by an odd number and x^2 - 9 is divisible by the same odd number.
Since x - 5 is the difference of two consecutive odd numbers, it is always divisible by 2. Therefore, for x - 5 to be divisible by an odd number, x must be an odd number.
Now, let's consider the values of x less than 301 that are odd.
Calculating the number of possible values:
From case 1, we found that there are no values of x for which (x + 3)(x - 3) has a common odd factor.
From case 2, we found that there are 17 values of x less than 301 that are odd.
Therefore, the total number of values of x that satisfy the given conditions is 17.
Conclusion:
Hence, the correct answer is option 'B' (17).
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DIRECTIONS for the question:Solve the following question and mark the best possible option.If the expression x2+ 9 and x + 5 are to have a common odd factor other than 1 such that x is a natural number less than 301, then how many values of x are possible.a)16b)17c)18d)15Correct answer is option 'B'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about DIRECTIONS for the question:Solve the following question and mark the best possible option.If the expression x2+ 9 and x + 5 are to have a common odd factor other than 1 such that x is a natural number less than 301, then how many values of x are possible.a)16b)17c)18d)15Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for DIRECTIONS for the question:Solve the following question and mark the best possible option.If the expression x2+ 9 and x + 5 are to have a common odd factor other than 1 such that x is a natural number less than 301, then how many values of x are possible.a)16b)17c)18d)15Correct answer is option 'B'. Can you explain this answer?.
DIRECTIONS for the question:Solve the following question and mark the best possible option.If the expression x2+ 9 and x + 5 are to have a common odd factor other than 1 such that x is a natural number less than 301, then how many values of x are possible.a)16b)17c)18d)15Correct answer is option 'B'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about DIRECTIONS for the question:Solve the following question and mark the best possible option.If the expression x2+ 9 and x + 5 are to have a common odd factor other than 1 such that x is a natural number less than 301, then how many values of x are possible.a)16b)17c)18d)15Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for DIRECTIONS for the question:Solve the following question and mark the best possible option.If the expression x2+ 9 and x + 5 are to have a common odd factor other than 1 such that x is a natural number less than 301, then how many values of x are possible.a)16b)17c)18d)15Correct answer is option 'B'. Can you explain this answer?.
Solutions for DIRECTIONS for the question:Solve the following question and mark the best possible option.If the expression x2+ 9 and x + 5 are to have a common odd factor other than 1 such that x is a natural number less than 301, then how many values of x are possible.a)16b)17c)18d)15Correct 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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