2
Part of 2005 May Olympiad
Problems(2)
Remainder and divison
Source: May Olympiad(Olimpiada de Mayo) 2005
2/21/2018
Gonçalo writes in a board four of the the following numbers , he can repeat numbers. Nicolas can realize the following operation: change one number of the board, by the remainder(in the division by ) of the product of others two numbers of the board. Nicolas wins if all the four numbers are equal, determine if Gonçalo can choose numbers such that Nicolas will never win.
combinatoricsnumber theory
6 consecutive autodivi integers wanted
Source: XI May Olympiad (Olimpiada de Mayo) 2005 L1 P2
9/22/2022
An integer is called autodivi if it is divisible by the two-digit number formed by its last two digits (tens and units). For example, is autodivi as it is divisible by , is autodivi since it is divisible by . Find consecutive integers that are autodivi and that have the digits of the units, tens and hundreds other than .
number theoryconsecutive