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 2 of 6: Rule out

n=3

Previous step Next step

n=3: \text{ every } k\ge3 \text{ has a prime power} > 2, \text{ so no set works}

Detailed analysis

For $n=3$ , Step 1 requires every prime power dividing $k$ to be at most $2$ , but any integer $k\ge3$ has a prime-power factor exceeding $2$ (e.g.\ $3,4,5,7,8,\dots$ ), violating the bound. So no set of $3$ consecutive integers has the required property.