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Least common multiple function !!!

Source: Romania TST 2015 Day 3 Problem 3

June 4, 2015

least common multiplefunctionRomanian TSTnumber theoryRIP

Problem Statement

If

k

and

n

are positive integers , and

k \leq n

, let

M(n,k)

denote the least common multiple of the numbers

n , n-1 , \ldots , n-k+1

.Let

f(n)

be the largest positive integer

k \leq n

such that

M(n,1)<M(n,2)<\ldots <M(n,k)

. Prove that : (a)

n

for all positive integers

n

. (b) If

N

is a positive integer , then

f(n) > N

for all but finitely many positive integers

n

.

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