Show that the sum of any \(100\) consecutive numbers is a multiple of \(50\) but not a multiple of \(100\).
Alice sums \(n\) consecutive numbers, not necessarily starting from \(1\), where \(n\) is a multiple of four. An example of such a sum is \(5+6+7+8\). Can this sum ever be odd?
Show that the difference between two consecutive square numbers is always odd.
Let \(n\) be a natural number and \(x=2n^2+n\). Prove that the sum of the square of the \(n+1\) consecutive integers starting at \(x\) is the sum of the square of the \(n\) consecutive integers starting at \(x+n+1\).
For example, when \(n=2\), we have \(10^2+11^2+12^2=13^2+14^2\)!
Find the contrapositive of the statement: “If in every school there is a class with at least \(20\) students, then there is a school with at least \(10\) students".
Show that if \(a\) and \(b\) are numbers, then \(a^2-b^2=(a-b)\times (a+b)\).
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.