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 5 of 5: Assemble the final answer

Previous step Next step

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

Detailed analysis

Step 1 handles $n=3,4,5$ directly, and Steps 2–4 handle every $n>5$ with two distinct valid largest elements $2p$ and $3p$ . Together, a valid set exists for every $n>3$ , and $n=4$ is the only case among all $n>3$ where exactly one valid set exists, since every other admissible $n$ has at least two.