Solution 3: Divisors and Primes

If is a prime number greater than , find the sum of the divisors of .

So, this turns out to be very straightforward.  The problem is that it appears to be intimidating to lots of students because they don’t have a firm grasp algebraically on the concept of a prime number.  And this, to me at least, seems like the major stumbling block on all of number theory, which is why it is such a fertile source of good contest problems.  You will never have a firm algebraic grasp of a prime number (ignoring of course the rather large subject of algebraic number theory), so it suddenly seems that all of the high school training in algebra is no good here.  Nevertheless, the main way to solve problems is to actually attempt to solve them instead of throwing up your hands and bathering in despair like a souless zombie from Left 4 Dead Two.

Problems 3 and 4: Divisors and Primes

Students who begin studying for math competitions have a tendency to be very competent at algebra, trig and anything calculus-like, mainly because that is what they have studied for years on end through school.  However, in geometry and number theory, especially anywhere in North America, my students, even the brighter ones, have quite a few more challenges ahead of them.  The lack of number theory or geometric repetition in their daily math routines often leaves the students frustrated and confused when forced to tackle anything that is outside their immediate sphere of knowledge.

To combat this of course, we need to expose the students to as many problems as possible, giving them time to solve them, but not so long that they abandon the subject.  Here are a few that I find interesting.  One easy, the other more challenging.

One: If is a prime number greater than , find the sum of the divisors of .

Two: If and are positive integers such that , show that is divisible by .

Solution 2: Large Numbers

Using the digits 1, 2, 3 and 4 just once, what is the largest number that may be formed?  Solution below the break.

Problem 2: Large Numbers

This is a problem that I will often show my junior classes, although on occasion it will also stump some of my more senior students.  Given the digits 1, 2, 3 and 4, what is the largest number that can be formed using each digit just once?

 

 

Solution 1: Fault Lines

We want to prove that all squares that are covered by tiles must a have a fault line.

Problem 1: Fault Lines

Here is something I discovered a few weeks ago and just couldn’t resist sharing with all my friends (hey, does this mean that you are also considered my friends, or does it mean that I’m actually just someone with really weird friends?).

Consider a 6×6 rectangle, and cover it with 2 x 1 dominoes or tiles.  A fault-line is defined as a line from one side of the square to the other that is not also cutting through a tile.  Prove that any covering of the square must have at least one fault-line.

Math Problems

For some reason that has completely eluded my conscious thought patterns, I wanted to take some time to put together 400 mathematical problems and their solutions.  These problems are all (in theory) solvable with ingenuity, perseverance and straightforward mathematics.  Mostly, the problems have come up as part of my day job, which is a high school math and computer science teacher.  However, please don’t hold that against me.  As much as possible, I will try and cite any sources for the problems that I post here, although on occasion, I just don’t know where they came from originally, and see them only as a question posed by a teacher in a neighbouring classroom.

Problems and solutions will originate from such fine pedigree as geometry, number theory, pigeons and their domiciles, computer science, games and inequalities.  Of course, given the nature of math problems, especially fine ones, this list is certainly not exhaustive.

Now, given that every person is more than just the sum of his or her parts, and this seems to be a very one-sided part of me (math ‘n stuff), I suppose that on occasion, I will also wax eloquently about other things, although realistically, who knows where this will go?