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 5: Invoke Bertrand's postulate

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In plain words

Bertrand's postulate is the classic tool for guaranteeing a prime in a range roughly half as wide as the range itself — exactly what is needed to place a prime inside a window of $n$ consecutive integers.

n>5: \exists \text{ prime } p \text{ with } \left\lfloor \tfrac n2 \right\rfloor < p < 2\left\lfloor \tfrac n2\right\rfloor, \text{ so } p<n\le 2p

Detailed analysis

Bertrand's postulate guarantees a prime strictly between $\lfloor n/2\rfloor$ and $2\lfloor n/2\rfloor$ ; rewriting the bounds shows this prime $p$ satisfies $p<n\le 2p$ .