4

Part of 2010 May Olympiad

Problems(2)

Source: May Olympiad(Olimpiada de Mayo) 2010

n

1<n<2010

, where we have a polygon with

2010

n

coins, we have to paint the vertices of this polygon with

n

colors and we've to put the

n

n

vertices of the polygon. In each second the coins will go to the neighbour vertex, going in the clockwise. Determine the values of

n

such that is possible paint and choose the initial position of the coins where in each second the

n

coins are in vertices of distinct colors

multiples of 13 wanted

Source: XVΙ May Olympiad (Olimpiada de Mayo) 2010 L1 P4

Find all natural numbers of

90

digits that are multiples of

13

and have the first

43

digits equal to each other and nonzero, the last

43

digits equal to each other, and the middle

4

2, 0, 1, 0

, in that order.

number theorymultiple