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 1 of 5: Check the small cases directly

Previous step Next step

n=3: \text{ no } m; \quad n=4: \text{ only } m=3 \ (\{3,4,5,6\}); \quad n=5: \ m=3,8

Detailed analysis

A direct search over small largest elements $m$ , writing the set as $\{m-n+1,\dots,m\}$ , confirms there is no valid $m$ for $n=3$ ; exactly one, $m=3$ , for $n=4$ ; and two values, $m=3$ and $m=8$ , for $n=5$ .