The Function g(x) = gcd(x, 10)

Date: 01/14/99 at 17:41:43
From: Omega Mejia
Subject: PRAXIS EXAM QUESTION

I'm going to be taking the Praxis exam in one week. I want to teach 
high school math. I hope you can help me with my question. The 
question that I have is from a Praxis sample test.  

Graph the function g(x) = gcd(10, x), where x is a positive integer.
I understand that the point (10,10) is on the graph, but when I look 
at the answer key I get confused. The graph has several points plotted 
on y = 1, 2, 5, and 10. I don't understand how they graphed this.

The next part to this question is: What is the range? Again the answer 
is 1, 2, 5, 10, but I don't know why.

The final part to this question is: If a positive integer x is chosen 
at random, assign a probability that g(x) = r for each r in the range 
of g(x). Justify your assignment.

Please consider my question. Thanks for your time.

Date: 01/19/99 at 15:58:01
From: Doctor Schwa
Subject: Re: PRAXIS EXAM QUESTION

Pick any integer value of x. What is the greatest common divisor of
x and 10? Well, it has to be a COMMON divisor, so it must be a divisor 
of 10. Thus, 1, 2, 5, and 10 are the only possible values. The possible 
values of the function are the range, {1, 2, 5, 10}.

To plot a few points, gcd(1,10) = 1, gcd(2, 10) = 2, since 2 goes into
both, gcd(3,10) = 1, gcd(4,10) = 2, gcd(5,10) = 5, gcd(6,10) = 2 ....

So the graph goes 1 2 1 2 5 2 1 2 1, and then gcd(10,10) = 10. Then it 
repeats again, 1 2 1 2 5 2 1 2 1, and then gcd(20,10) = 10 again!

Now what's the probability? In the sequence, 10 occurs once out of each 
ten numbers (the multiples of 10), 5 occurs once out of each ten 
numbers (the multiples of 5 that aren't also multiples of 10, or in 
other words, numbers ending in 5), 2 occurs 4 times (numbers ending in 
2, 4, 6, 8), and 1 occurs 4 times (numbers ending in 1, 3, 7, 9).

So I would assign a probability of 1/10 to g = 10, 1/10 to g = 5, and 
4/10 to each of g = 2 and g = 1.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/