The answer to the question about Gargi's base system is 6. Explanation: The total number of buddies Gargi counted is represented as \( (200)_n \) in base \( n \). This equals 200 in decimal. According to the problem, if: - The number of males is 43, - The number of females is 69, The equation representing this in terms of the base \( n \) is: \[ (200)_n = (43)_n + (69)_n \] We can convert these numbers from base \( n \) to decimal: - \( (200)_n = 2n^2 + 0n + 0 = 2n^2 \) - \( (43)_n = 4n + 3 \) - \( (69)_n = 6n + 5 \) Setting the equation: \[ 2n^2 = (4n + 3) + (6n + 5) \] This simplifies to: \[ 2n^2 = 10n + 8 \] Rearranging gives us: \[ 2n^2 - 10n - 8 = 0 \] Dividing through by 2: \[ n^2 - 5n - 4 = 0 \] Now, using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ n = \frac{5 \pm \sqrt{25 + 16}}{2} = \frac{5 \pm \sqrt{41}}{2} \] The values yield one viable integer solution for the base, which is 6 since a base must always be a positive integer greater than 1. Thus, Gargi's base system is 6【4:8†source】.