Integers, Prime Numbers And Prime Factorizations I would like to begin today with a few simple definitions of terms that appear in number theory problems on the ACT , GRE and SAT math subject tests. Definitions The integers are the counting numbers together with their negatives. …,-4, -3, -2, -1, 0, 1, 2, 3, 4,… The positive integers consist of the positive numbers from the above list. 1, 2, 3, 4,… Next we have prime numbers. A prime number is a positive integer that has exactly two factors (1 and itself). Here is a list of the first few primes: 2, 3, 5, 7, 11, 13, 17, 19, 23,… Note that 1 is not prime. It has only one factor. A little trick: Here is a quick trick for determining if a large number is prime: take the square root of the integer and check if the integer is divisible by each prime up to this square root. If not, the number is prime. For example let’s try to figure out if 3001 is a prime number. Note that when we take the square root of 3001 in our calculator we get approximately 54.8. Now with our calculators we divide 3001 by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, and 53 (all the prime numbers below 54.8). Since none of these are integers, 3001 is prime. The Fundamental Theorem of Arithmetic The fundamental theorem of arithmetic says “every integer greater than 1 can be written “uniquely” as a product of primes.” The word “uniquely” is written in quotes because prime factorizations are only unique if we agree to write the primes in increasing order. For example, 6 can be written as 2 · 3 or as 3 · 2. But these two factorizations are the same except that we changed the order of the factors. To make things as simple as possible we always agree to use the canonical representation. The word “canonical” is just a fancy name for “natural,” and the most natural way to write a prime factorization is in increasing order of primes. So the canonical representation of 6 is 2 · 3. As another example, the canonical representation of 18 is 2 · 3 · 3. We can tidy this up a bit by rewriting 3 · 3 as 32. So the canonical representation of 18 is 2 · 32. If you are new to factoring, you may find it helpful to draw a factor tree. For example here is a factor tree for 18: To draw this tree we started by writing 18 as the product 2 · 9. We put a box around 2 because 2 is prime, and does not need to be factored anymore. We then proceeded to factor 9 as 3 · 3. We put a box around each 3 because 3 is prime. We now see that we are done, and the prime factorization can be found by multiplying all of the boxed numbers together. Remember that we will usually want the canonical representation, so write the final product in increasing order of primes. By the Fundamental Theorem of Arithmetic above it does not matter how we factor the number – we will always get the same canonical form. For example, here is a different factor tree for 18: For practice, why don’t you try to draw a factor tree for 6137? Note that this is much more challenging than any number you will have to factor on a standardized test. I’ll have the solution for you tomorrow, so we can compare notes then. Remember, I am not looking for artistic merit – just make your factors clear in your drawing. How many prime numbers are there? This is going off on a bit of a tangent, but there are an infinite number of prime numbers. This was first proved by the ancient Greek mathematician Euclid. Interestingly, the largest prime number found so far is 257,885,161 – 1. That’s a number with 17 million digits! I dare you to find the next one greater than that. In the meantime, if you want to learn mathematical strategies to efficiently answer math questions on standardized tests, I would suggest that you take a look at the Get 800 collection of test prep books. Click on the picture below for more information about these books. Comments comments