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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?
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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.
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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).
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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?
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