Sunday, March 15, 2009
Those "Function" Questions on the SATs - Practice, Tips
PLS NOTE THE EDIT TO THE PROBLEM BELOW. THE ORIGINAL WORDING WAS INACCURATE.
The following is not a classic function question even though it uses function notation. This is an original problem I wrote but it is the kind of question that might appear. The level of difficulty would be medium. The math content is middle school level but the wording and notation are the challenge for most students. Beyond preparing students for a test like the SATs, my strong belief is that such questions should be included in textbooks from middle school on (even with that function notation!). This question reviews basic math concepts (primes, factors, gcf) and can also be used as a springboard for discussion of the concept of "relatively prime", Euler's phi function, π(x) and other number-theoretic topics.
Note: The "For example" hint may or may not be included in the question. It certainly makes the notational issue less formidable.
If n is a positive integer greater than 1, then the sets F(n) and P(n) consist only of positive integers and are defined as follows:
A positive integer, k, belongs to the set F(n) if k ≤ n and the greatest common factor of k and n equals 1.
A positive integer, k, belongs to the set P(n) if k ≤ n and k is prime.
For example, F(6) contains the numbers 1 and 5 and therefore has two elements. P(6) contains the numbers 2, 3 and 5 and therefore has three elements.
What is the ratio of the number of elements in F(20) to the number of elements in P(20)?
Click Read more below to see answer (suggested solution will be posted later).
Answer: 1
Explanation: Not yet...
Posted by Dave Marain at 9:19 AM 6 comments
Labels: divisibility, functional notation, functions, gcf, more, primes, SAT-type problems