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 6: Resolve the boundary case

n=4

Previous step Next step

n=4:\quad \{3,4,5,6\} \text{ is the only set (}6=\mathrm{lcm}(3,4,5)\text{)}

Detailed analysis

For $n=4$ , Step 4's construction is unavailable, and a direct check of small largest elements shows $k=6$ , the set $\{3,4,5,6\}$ , is the only value satisfying the Step 1 bound, since every integer other than $6$ that is at least $4$ has a prime-power factor exceeding $3$ .