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p-expansions of m and n

Source: Turkey TST 1999 - P1

July 2, 2012

number theory proposednumber theory

Problem Statement

Let

m \leq n

be positive integers and

p

be a prime. Let

p-

expansions of

m

and

n

be

m = a_0 + a_1p + \dots + a_rp^r

n = b_0 + b_1p + \dots + b_sp^s

respectively, where

a_r, b_s \neq 0

, for all

i \in \{0,1,\dots,r\}

and for all

j \in \{0,1,\dots,s\}

, we have

0 \leq a_i, b_j \leq p-1

. If

a_i \leq b_i

for all

i \in \{0,1,\dots,r\}

, we write

m \prec_p n

. Prove that

p \nmid {{n}\choose{m}} \Leftrightarrow m \prec_p n

.

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