Show that if \(a\) and \(b\) are numbers, then \(a^2-b^2=(a-b)\times (a+b)\).
One of the most important tools in maths is the Pigeonhole Principle.
You may have already met it before, but if not, let’s recap it quickly.
Simply put: the Pigeonhole Principle states that if you have \(n\) pigeons (or objects) that you want to
place into a number of pigeonholes (or containers) that is strictly
smaller than \(n\), e.g: \(10\) pigeons but only \(9\) pigeonholes, then there will be a
pigeonhole with at least two pigeons. Today we will see how this
principle can be used to solve problems about numbers and their
divisibility properties.
Before we get started, we need to recap a very important concept: if we
have two numbers, say \(a\) and \(b\), we can divide \(a\) by \(b\), and we will obtain a quotient
\(q\) and a remainder \(r\), and write \[a=q\times b + r\] for example: if we
divide \(9\) by \(4\), we can write \(9=2\times 4 + 1\), i.e: the quotient will
be \(2\) and the remainder will be
\(1\).
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 \(6\) numbers, 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 \(11\) 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 \(12\).
There are six kids in the math circle. Each kid has their own seat, and they always sit in the same one. One day, however, the head tutor decided to rearrange the seating, and it turned out that every kid ended up in a different seat from their usual one. In how many ways can the head tutor do this?
Seven students are standing on a straight line, one after the other. Three of the students, let’s call them \(A,B,\) and \(C\) behave badly and can’t be next to each other. For example: \(\star \star AB\star \star C\) and \(\star ABC\star \star \star\) are invalid arrangements, where the star denotes any other student. However, \(A\star B\star \star \star C\) is an example of a valid arrangement. How many valid arrangements are there?