Expert: Scott A Wilson - 5/9/2009

Question
QUESTION: How do you prove: 2n+1 and n+1 are relatively prime?

ANSWER: The only common factor between 2n+1 and n+1 is 1,
therefore they are relatively prime.


If the numbers were 2n+2 and n+1, then the factors of 2n+2 would be
2(n+1).  In this case, n+1 would go into both, so for n positive, the numbers would not be prime.

If the numbers were 3n+3 and 6n+9, the factors would be
3(n+1) and 3(2n+3).  Here, 3 would be a common factor.


---------- FOLLOW-UP ----------

QUESTION: Ok Scott, this is a follow up question on this problem.
Let's suppose n is even.  Then, by def. of even,n=2k,where k is any positive integer.  Now, 2n+1=2(2k)+1=4k+1. n+1 = 2k+1.  What if k is 4? Then 2k+1 would be 9 and 9 is divisible by 3.  

Then I took the scenario that n was odd.  There I find the same thing. So my conclusion is that the numbers are not relatively prime.
What do you think?

Evelyn

ANSWER: If k were 4, that would make the second number 9.
The first number, though, is said to be 4k+1.
That makes that number 17, which is prime by itself.  
That definitey makes the numbers relatively prime.

It does no matter if one of the numbers is prime or not.
The problem states that they have to be relatively prime.

Put in n as 7 and the numbers are 2*7 + 1 = 15 and 7+1 = 8.
The number 15 can be seen to be 3*5.  The number 8 is 2*2*2.
Neither one is prime, but they are relatively prime since they have no factors in common.

Or take 32.  2*32 + 1 = 65 = 5*13.  32+1 = 33 = 3*11.
Neither of these is a prime number, but they are relatively prime since they have no factors in common.


---------- FOLLOW-UP ----------

QUESTION: Great analysis.  Last question in reference to the same:
Did I, however, proceed correctly by assuming n=even and later n=odd?  Or would you have done it exactly how you just explained it?  When we prove can we actually put "number" scenarios?
Thanks Scotto.... now I understand what "relatively" prime means.

Evelyn

Answer
I was thinking about it for awhile and it made sense to consider n as even and as odd in the way I was approaching it, but when an easier way is found, the other is not looked at anymore.

Like when you know that the hypoteneuse of a triangle's length c is given by c² = b² + a², that's what you do.  Before there were calculaters (like back when my older brother was a kid), really good kids in math were taught the method to find the squareroot by hand.
That's where I learned my math from.  I was eight years younger than him and asked him lots of math questions, so I didn't really learn much until I got to calculus my senior year.

That's it - we'll blame my interest in my big brother for all this math I know!  That's right, but I sure love it.

Have you ever gone to sleep doing math for fun?
Has it ever entered your dreams?
I sometimes dream about math, like people asking me about how far it is, and I tell them to compute the squares of the sides, add them up, then find the squareroot.  And then there's one about something like computing how far off shore we were by knowing the angle and the height of what the person was on shore.  That sounds strange, but the person was somewhere around 53 feet tall.  It was also strange that we knew how tall they were.  Don't ask - its just a dream.

Even when I was in the hospital recovering from a 3 and half week coma, I could still do math.  I couldn't walk and couldn't talk, but I could still do math.  It came back to me a little after I was able to open my eyes.  It must be something like an automatic reflex for me.  While I'm breathing, I sometimes think about the energy used in moving however much air in and out.  Or maybe the change in temperature from knowing the body temperature and the air temperature.  It varies.  The colder it is outside, the more of a change is generated in increasing the termperature.

How about math while you're walking... how many steps are there in a mile for the average person?  Around 2000 or so?  Don't know.
Or how many times do you breathe?  In a day?  In a month?
I noticed I do it far less than the average person.