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Divisibility of Binomial Differences

Source: SAMO 2016 Q6

September 24, 2016

number theory

Problem Statement

Let

k

and

m

be integers with

1 < k < m

. For a positive integer

i

, let

L_i

be the least common multiple of

1,2,\ldots,i

. Prove that

k

is a divisor of

L_i \cdot [\binom{m}{i} - \binom{m-k}{i}]

for all

i \geq 1

. [Here,

\binom{n}{i} = \frac{n!}{i!(n-i)!}

denotes a binomial coefficient. Note that

\binom{n}{i} = 0

if

n < i

.]

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