Are there any two-digit numbers which are the product of their digits?
Show that given any three numbers, at least two of them will have the same parity. Recall that the parity of a number is whether it is odd or even.
Show that given any \(6\) whole numbers - not necessarily consecutive - at least two of them will have the same remainder when divided by \(5\).
Show that given any \(3\) numbers, there will be two of them so that their difference is an even number.
Show that given \(11\) whole numbers, there will be at least \(2\) numbers whose difference ends in a zero.
Three whole numbers are marked on a number line. Show that for two of these marked numbers, the point halfway between them is also a whole number.
Show that among any \(51\) whole numbers, all at most \(100\), there must be two that share no prime factors. For example, \(7\) and \(8\) share no prime factors, and the same is true for \(11\) and \(12\).
Show that if any \(12\) two-digit numbers are given, you can always choose two of them such that their difference is of the form \(AA\) where \(A\) is some digit from \(0\) to \(9\).
You are given fifty-one whole numbers. Assume that the square of one number equals the sum of the squares of all the other fifty numbers. Prove that among these fifty-one numbers, there must be an even one.
Show that if \(k\) is a positive whole number, then the decimal expansion of \(1/k\) either has a finite number of decimal places or eventually repeats. For example, \[\frac{1}{5} = 0.2 \qquad\text{or}\qquad \frac{1}{17} = 0.\underbrace{0588235294117647}_{} \underbrace{0588235294117647}_{}\ldots\]