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XXX OBM

Brazil counting and probability

Problem

A positive integer is dapper if at least one of its multiples begins with

2008

. For example,

7

is dapper because

200858

is a multiple of

7

and begins with

2008

. Observe that

200858 = 28694 \times 7

. Prove that every positive integer is dapper.

Solution

Let

n

be any positive integer. Choose

k

to be an integer greater than the number of digits of

n

. So the interval

[2008 \cdot 1 0^{k}, 2009 \cdot 1 0^{k} [

, which contains

1 0^{k} > n

integer consecutive numbers, has a multiple of

n

. So every positive integer

n

is dapper.

Techniques

Pigeonhole principleOther