Solution: To determine the probability that the product of three randomly chosen integers between 0 and 100 is divisible by 5, we use complementary probability. First, note that an integer is divisible by 5 if it ends in 0 or 5. Among the integers from 0 to 100, there are 21 multiples of 5 (i.e., 0, 5, 10, ..., 100). Thus, the probability that a single randomly chosen integer is **not** divisible by 5 is:

["Title: Calculating the Probability That the Product of Three Random Integers (0 to 100) Is Divisible by 5 Using Complementary Probability", "Meta Description: Learn how to calculate the probability that the product of three randomly chosen integers between 0 and 100 is divisible by 5 using complementary probability—start by analyzing when a number isn’t divisible by 5.", "---", "When selecting three integers randomly between 0 and 100, a key question arises: What is the probability that their product is divisible by 5? At first glance, this might seem complex, involving permutations and multiple combinations. However, a powerful mathematical principle simplifies this problem: complementary probability.", "### Why Use Complementary Probability?\nThe product of the three numbers is divisible by 5 if at least one of the numbers is divisible by 5. Instead of calculating probabilities for one, two, or all three numbers divisible by 5, we compute the complementary case—when none of the three numbers is divisible by 5—and subtract it from 1.", "This approach reduces complexity and ensures accuracy.", "### Step 1: Determine When a Number Isn’t Divisible by 5\nAn integer between 0 and 100 is divisible by 5 if its last digit is 0 or 5. From 0 to 100 inclusive:\n- The multiples of 5 are: 0, 5, 10, ..., 100\n- Count: ( \frac{100 - 0}{5} + 1 = 21 ) numbers", "So, out of 101 total integers (0 to 100), 21 are divisible by 5.", "Therefore, 21 out of 101 integers are divisible by 5, but 100 out of 101 integers are NOT divisible by 5.", "Thus, the probability that a single randomly chosen integer from 0 to 100 is not divisible by 5 is:\n[\n\frac{100}{101}\n]", "### Step 2: Compute the Complementary Event\nThe complementary event to “the product is divisible by 5” is “the product is not divisible by 5,” which happens only if none of the three chosen numbers is divisible by 5.", "Since selections are independent:\n[\nP(\ ext{none divisible by 5}) = \left( \frac{100}{101} \right)^3\n]", "### Step 3: Apply Complementary Probability\nNow, the desired probability is:\n[\nP(\ ext{product divisible by 5}) = 1 - P(\ ext{none divisible by 5}) = 1 - \left( \frac{100}{101} \right)^3\n]", "Calculating:\n[\n\left( \frac{100}{101} \right)^3 \approx \frac{1,000,000}{1,030,301} \approx 0.9705\n]\n[\n1 - 0.9705 = 0.0295 \quad \ ext{(approximately)}\n]", "So, the probability is roughly 2.95%.", "But let’s express it precisely:\n[\nP = 1 - \left( \frac{100}{101} \right)^3 = 1 - \frac{1,000,000}{1,030,301} = \frac{30,301}{1,030,301}\n]", "### Real-World Implications\nThis result shows that it’s relatively unlikely—less than 3%—that three random integers between 0 and 100 will multiply to a number divisible by 5. This insight is valuable in number theory, probability modeling, and risk assessment scenarios involving random selection.", "### Summary\nBy using complementary probability, we transformed a complex “at least one” scenario into a simple “all not” calculation—demonstrating how strategic event analysis streamlines probability problems. The final answer is:\n[\n\boxed{1 - \left( \frac{100}{101} \right)^3}\n]\nor approximately 2.95%.", "---", "Keywords: probability divisible by 5, three integers 0 to 100, complementary probability, mathematical modeling, number theory probability, random selection, complementary event, statistical analysis."]