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Home » 5 clusters: not valid since $12 \div 5 = 2.4$, not an integer - Appfinity Technologies

5 clusters: not valid since $12 \div 5 = 2.4$, not an integer - Appfinity Technologies

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 2Question: What is the remainder when the sum of the first 100 even numbers is divided by 7?

📌 Solution: The first 100 even numbers form an arithmetic sequence: $2 + 4 + 6 + \dots + 200$. The sum is $S = \frac{100}{2} \times (2 + 200) = 50 \times 202 = 10100$. To find $10100 \mod 7$, divide 10100 by 7: $7 \times 1442 = 10094$, so the remainder is $10100 - 10094 = 6$. $\boxed{6}$

📌 Question: What is the largest integer that must divide the product of any three consecutive integers representing the number of trees planted in three consecutive years?

📌 Solution: Any three consecutive integers include at least one multiple of 2 and one multiple of 3. Thus, their product is divisible by $2 \times 3 = 6$. For example, $1 \times 2 \times 3 = 6$, $2 \times 3 \times 4 = 24$, and $3 \times 4 \times 5 = 60$. The greatest common divisor of all such products is 6. $\boxed{6}$

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