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with first term $ a = 3 $, common difference $ d = 5 $, and last term $ l = 98 $.
Use the formula for the $ n $-th term:
Question: What is the greatest common factor of 78 and 117?
The common prime factors are $ 3 $ and $ 13 $. Taking the smallest powers:
Question: Find the least common multiple of 24 and 36.
Take the highest powers of all primes:
Question: A historian of science is organizing a roundtable discussion with 7 participants, including 2 experts on Newtonian physics and 5 historians of technology. If the seating is circular and the 2 Newtonian physics experts must sit together, how many distinct seating arrangements are possible?
Solution: Since the seating is circular and the 2 Newtonian experts must sit together, we treat them as a single block. This reduces the problem to arranging 6 units (the block and 5 historians) in a circle. The number of circular permutations of $n$ objects is $(n-1)!$, so we fix the block and arrange the 5 historians around it, giving $(6 - 1)! = 5!$ arrangements. Within the block, the 2 Newtonian experts can be arranged in $2!$ ways. Thus, the total number of arrangements is:
Question: A geographer is analyzing elevation data from 8 different satellite images. Each image is categorized as either high elevation (H) or low elevation (L). If exactly 3
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