Final answer:
The 150th shopper will be the first to receive all five gifts since 150 is the least common multiple of 5, 10, 25, and 75.
Step-by-step explanation:
To determine which shopper will be the first to receive all five gifts during the store's 100th anniversary, we need to find the least common multiple (LCM) of the numbers that represent each giveaway interval: pin (every person), additional pin (every 5th person), hand sanitizer (every 10th person), umbrella (every 25th person), and free dinner (every 75th person). The LCM of 5, 10, 25, and 75 is 150. This is because 150 is the smallest number that is divisible by all of these numbers. Hence, the 150th shopper will be the first to receive the pin (for just entering), a second pin (5th person), hand sanitizer (10th person), an umbrella (25th person), and a free dinner (75th person).