MathDB

2

Part of 2018 Saudi Arabia GMO TST

Problems(2)

exists integer n such that p | n^3 - 3n + 1 where p a prime of form 9k + 1

Source: 2018 Saudi Arabia GMO TST I p2

7/31/2020
Let pp be a prime number of the form 9k+19k + 1. Show that there exists an integer n such that p∣n3−3n+1p | n^3 - 3n + 1.
number theorydividesdivisibleprime
similar integer, swapping 2 digits

Source: 2018 Saudi Arabia GMO TST II p2

7/31/2020
Two positive integers mm and nn are called similar if one of them can be obtained from the other one by swapping two digits (note that a 00-digit cannot be swapped with the leading digit). Find the greatest integer NN such that N is divisible by 1313 and any number similar to NN is not divisible by 1313.
number theoryDigitsdivisible