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 3 of 5: First working set for

n>5

Previous step Next step

m=2p: \quad 2p \mid \mathrm{lcm}(2p-1,\dots,2p-n+1)

Detailed analysis

Since $n-1\ge p$ , the window $\{m-1,\dots,m-n+1\}$ with $m=2p$ contains a multiple of $p$ (hence, together with the multiple of $2$ also present, a multiple of $2p$ ), of $3$ , and of $2$ , so their lcm is divisible by $2p$ , making $m=2p$ a valid largest element.