Divisibility constraint: 7x+96 divisible by x
Question
Let x be a positive integer such that 7x + 96 is divisible by x. How many values of x are possible?
Options
10
11
12
Infinitely many
Explanation
Express the divisibility statement as an algebraic fraction: 7x + 96/x = 7x/x + 96/x = 7 + 96/x.
For this expression to evaluate to an integer outcome, the variable ⟨MATH⟩x⟨/MATH⟩ must be a perfect factor of 96. Find the total factor count by expanding 96 into its prime factorization form: 96 = 32 × 3 = 2^5 × 3^1.
Apply the standard exponent factor formula (a_1 + 1)(a_2 + 1) \dots: Total factors = (5 + 1) × (1 + 1) = 6 × 2 = 12. Since x must be a positive integer, there are exactly 12 valid values.
Answer: (c).
Question details
Year
2023
Paper
CSAT
Question
Q25
Section
Numerical Ability
Sub-topic
Factors & Divisors
Type
Number theory
Difficulty
Medium
Source hint
Number theory
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