1
Part of 2004 Bundeswettbewerb Mathematik
Problems(2)
Modulo rulez
Source: BWM 2004, 1st round, problem 1
9/3/2004
At the beginning of a game, I write the numbers , , ..., onto a desk. A move consists of
- selecting some numbers standing on the desk;
- calculating the rest of the sum of these numbers under division by and writing this rest onto the desk;
- deleting the selected numbers.
In such a game, after a number of moves, only two numbers remained on the desk. One of them was . What was the other one?
Easy number theory from the bwm
Source: BWM 2004, 2nd round, problem 1
9/1/2004
Let be a positive integer. A natural number is called -typical if each divisor of leaves the remainder when being divided by .
Prove:
a) If the number of all divisors of a positive integer (including the divisors and ) is -typical, then is the -th power of an integer.
b) If , then the converse of the assertion a) is not true.
modular arithmeticnumber theory proposednumber theory