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?

February 22, 2026

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?

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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 $10100 - 10094 = 6$. $\boxed{6}$
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. $\boxed{6}$
Question: What two-digit positive integer is one less than a multiple of 5 and also one less than a multiple of 7?
Solution: Let $x$ be the integer. Then $x \equiv -1 \pmod{5}$ and $x \equiv -1 \pmod{7}$. This implies $x + 1$ is divisible by both 5 and 7, so $x + 1 = \text{LCM}(5, 7) = 35$. Thus, $x = 34$. $\boxed{34}$Question: A retired engineer designing a robotic arm observes that the sum of two gear rotation speeds $ a $ and $ b $ is 10, and the sum of their squares is 58. What is $ a^3 + b^3 $?

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