Problem 4

(a) For which integers

n>2

does there exist a set of

n

consecutive positive integers such that the largest number in the set divides the least common multiple of the remaining

n-1

numbers? (b) For which integers

n>2

is there exactly one such set?

Step 6 of 6: Conclude both parts

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\text{(a) all } n>3 \text{ admit a set; (b) only } n=4 \text{ admits exactly one}

Detailed analysis

Combining Steps 2–5: sets exist exactly for $n>3$ (never for $n=3$ ), and among those, $n=4$ is the only case where the set is unique, since every $n>4$ has, at least, the two distinct constructions of Steps 3 and 4.