Study the following information carefully to answer these questions: A young girl Roopa leaves home with $x$ flowers and goes to the bank of a nearby river. On the bank of the river, there are four places of worship, standing in a row. She dips all the $x$ flowers into the river, the number of flowers doubles. Then, she enters the first place of worship and offers $y$ flowers to the deity. She dips the remaining flowers into the river, and again the number of flowers doubles. She goes to the second place of worship and offers $y$ flowers to the deity. She dips the remaining flowers into the river and again the number of flowers doubles. She goes to the third place of worship and offers $y$ flowers to the deity. She dips the remaining flowers into the river and again the number of flowers doubles. She goes to the fourth place of worship and offers $y$ flowers to the deity. Now she is left with no flowers in hand. The minimum number of flowers with which Roopa leaves home is

### Concept & Logic This question builds directly on the ratio derived from the doubling sequence. We know that after 4 cycles of doubling the current amount and subtracting $y$, the final amount is zero. This yields a specific linear Diophantine equation where we must find the smallest positive integer solution for $x$. ### Step-by-Step Solution * **Given:** * The iterative process results in the equation: $16x - 15y = 0$ * $x$ represents the initial number of flowers she leaves home with. * $y$ represents the flowers offered to each deity. * **Calculation:** 1. Rewrite the equation to isolate the ratio of $x$ to $y$: $$16x = 15y$$ $$\frac{x}{y} = \frac{15}{16}$$ 2. For $x$ and $y$ to be whole numbers (you cannot have fractions of a flower), they must be multiples of the numerator and denominator in their simplest form. Let $k$ be a positive integer. $$x = 15k$$ $$y = 16k$$ 3. The question asks for the minimum number of flowers Roopa leaves home with, which is the minimum positive value for $x$. 4. Set $k = 1$ to find the smallest positive integer. $$x = 15(1) = 15$$ ### Exam Strategy & Shortcut Using the generalized formula for this puzzle type: $2^n(x) = (2^n - 1)y$. With $n=4$ temples, this becomes $16x = 15y$. To find the minimum $x$, simply look at the coefficient attached to $y$. Because 16 and 15 share no common factors, the smallest positive integer for $x$ is exactly 15. ### Common Pitfall A frequent mistake is mixing up which variable corresponds to which value. Students might incorrectly choose 16 (the minimum for $y$) instead of 15 (the minimum for $x$). Always double-check your initial definitions: $x$ was defined as the flowers she "leaves home with." ### Final Answer Therefore, the correct answer is 15.