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 4 of 6: Second construction for

n>4

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In plain words

Having two different constructions immediately shows uniqueness fails for every $n>4$ : whenever you can build one qualifying set, in this range you can build a second one too.

n>4: \quad k'=(n-2)(n-3)=\mathrm{lcm}(n-2,n-3) \text{ also works}

Detailed analysis

By the same reasoning as Step 3 applied one step earlier, $k'=(n-2)(n-3)$ gives another valid set of $n$ consecutive integers, distinct from the one in Step 3 whenever $n>4$ (so that $k'\ne k$ ).