Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.
Prove that for any odd natural number, \(a\), there exists a natural number, \(b\), such that \(2^b - 1\) is divisible by \(a\).
What has a greater value: \(300!\) or \(100^{300}\)?
Prove that, for any integer \(n\), among the numbers \(n, n + 1, n + 2, \dots , n + 9\) there is at least one number that is mutually prime with the other nine numbers.
Every integer from 1 to 64 is written in an \(8 \times 8\) table. Prove that no matter the order these numbers are written, there are two adjacent numbers which are at least \(5\) apart. (Numbers in cells which share a side are called adjacent).
Prove that in any group of 2001 whole numbers there will be two whose difference is divisible by 2000.
Prove that amongst the numbers of the form \[19991999\dots 19990\dots 0\] – that is 1999 a number of times, followed by a number of 0s – there will be at least one divisible by 2001.
Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.
a) Prove that within any 6 whole numbers there will be two that have a difference between them that is a multiple of 5.
b) Will this statement remain true if instead of the difference we considered the total?
Will the quotient or the remainder change if a divided number and the divisor are increased by 3 times?