Let x be a positive integer such that 7x + 96 is divisible by x . How many values of x are possible?
- A10
- B11
- C12Correct
- DInfinitely many
Explanation
To solve this, we analyze the expression (7x + 96) / x. This can be separated into two parts: (7x / x) + (96 / x).
Simplifying the first part gives 7 + (96 / x).
For the entire expression to result in an integer, 96 must be divisible by x. This means x must be a factor of 96.
To find the number of factors of 96, we first find its prime factorization: 96 = 2 raised to the power of 5 multiplied by 3 raised to the power of 1.
The total number of factors is calculated by adding 1 to each exponent and multiplying them together: (5 + 1) * (1 + 1) = 6 * 2 = 12.
Since there are 12 factors of 96, there are 12 possible values for x. Therefore, the correct answer is C.
Related questions
More UPSC Prelims practice from the same subject and topic.
- Prelims 2023CSATQuantitative Aptitude
What is the remainder when 85 × 87 × 89 × 91 × 95 × 96 is divided by 100?
- Prelims 2023CSATQuantitative Aptitude
What is the unit digit in the expansion of (57242)^(9×7×5×3×1)?
- Prelims 2023CSATQuantitative Aptitude
If ABC and DEF are both 3-digit numbers such that A, B, C, D, E and F are distinct non-zero digits such that ABC + DEF = 1111 , then what is the value of A + B + C + D + E + F ?
- Prelims 2023CSATQuantitative Aptitude
D is a 3-digit number such that the ratio of the number to the sum of its digits is least. What is the difference between the digit at the hundred's place and the digit at the unit's place of D?
- Prelims 2023CSATQuantitative Aptitude
Three of the five positive integers p, q, r, s, t are even and two of them are odd (not necessarily in order). Consider the following: 1. p + q + r - s - t is definitely even. 2. 2p + q + 2r - 2s + t …
- Prelims 2023CSATQuantitative Aptitude
Consider the following in respect of prime number p and composite number c. 1. p + c{p - c} can be even. 2. 2p + c can be odd. 3. pc can be odd. Which of the statements given above are correct?