The explanation revolves around the Pigeonhole Principle and the concept of remainders when dividing by 8.75. Here's a step-by-step breakdown of the reasoning:

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Key Setup

1. Total Driving Time Boundaries:

   • Brad drives for 15 minutes (0.25 hours) or 30 minutes (0.5 hours) each day.
   • Over 35 days, the total driving time cannot exceed 15 hours.

2. Cumulative Driving Times:

   • Define $S_n$ as the total driving time from the first day to the $n$-th day.
   • For example, $S_1$ = driving time on Day 1, $S_2 = S_1 + \text{Day 2 driving time}$, and so on.

3. Objective:

   • Prove that there exists a period of consecutive days during which the total driving time equals exactly 8.75 hours.

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Using Modular Arithmetic

The critical observation is to consider the cumulative sums $S_1, S_2, \dots, S_{35}$ mod 8.75 (i.e., the remainder when dividing each cumulative sum by 8.75). This creates 35 cumulative sums and 35 possible remainders within the range $[0, 8.75)$.

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Case Analysis

Case 1: Two Sums Have the Same Remainder Modulo 8.75

• If two cumulative sums $S_i$ and $S_j$ (where $i < j$) have the same remainder when divided by 8.75, then: $S_i \equiv S_j \pmod{8.75}$ Subtracting these: $S_j - S_i = k \cdot 8.75$ Since the total driving time is at most 15 hours, the only feasible value of $k$ is 1 (as $2 \cdot 8.75 = 17.5$, which exceeds 15 hours). Thus: $S_j - S_i = 8.75$ This means that the driving time from Day $i+1$ to Day $j$ is exactly 8.75 hours.

Case 2: No Two Sums Have the Same Remainder Modulo 8.75

• If all 35 cumulative sums have unique remainders modulo 8.75, then one of these remainders must be 0 (since there are 35 possible sums and 35 possible remainders, including 0).
• A remainder of 0 implies: $S_i = 8.75 \cdot k$ For $k = 1$, $S_i = 8.75$, which directly shows that the cumulative driving time from Day 1 to Day $i$ is exactly 8.75 hours.

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Why This Covers All Possibilities

• Either two sums share a remainder modulo 8.75 (Case 1), or all sums have unique remainders (Case 2). In both cases, we demonstrate that there exists a period where the driving time totals 8.75 hours.

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Final Remark

The Pigeonhole Principle ensures the logic is exhaustive:

• There are 35 cumulative sums and 35 remainders, meaning at least one of the above cases must occur. In both cases, we achieve the desired result.

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