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 5: Second working set for

n>5

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m=3p: \quad 3p \mid \mathrm{lcm}(3p-1,\dots,3p-n+1)

Detailed analysis

By the same argument with $m=3p$ , the window again contains multiples of $p$ , of $2$ , and of $3$ among the preceding $n-1$ numbers, so $3p$ also divides the lcm of the rest, giving a second, distinct valid largest element since $2p\ne 3p$ .