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Solution: The sequence of pollinators per day is an arithmetic sequence with first term $ a = 1 $ and common difference $ d = 3 $. The formula for the $ n $-th term of an arithmetic sequence is:
Question: A mechanical engineer is designing a set of non-overlapping rectangular storage compartments for a surgical robot. If the robot has a total floor area of 60 square inches and each compartment must be a rectangle larger than 5 square inches but smaller than 15 square inches, what is the minimum number of such rectangles needed to exactly cover the area?
Solution: To minimize the number of rectangles, we should maximize the area of each rectangle without exceeding 15 square inches. The largest area per rectangle is 14 square inches (a possible value in the allowed range). Then we divide the total area by this value:
Since the number of rectangles must be a whole number and the rectangles must exactly cover the area without overlap, we round up to the next whole number:
Question: What is the probability that a randomly selected positive integer less than or equal to 50 is a divisor of 60?
Solution: First, list all positive divisors of 60. The prime factorization of 60 is:
However, only those less than or equal to 50 are valid. Removing 60:
There are 50 positive integers less than or equal to 50. So the probability is:
Solution: To find the smallest prime factor of 143, we test divisibility by small primes:
Both 11 and 13 are primes, and the smallest is: