algebra

[Guest]

What is the least positive interger n such that 3n - n3 is an even 4 digit #.

 

[Gene]

You can start with
3N-N^3=1000
to get the smalles possible N that gives any four digit number.

BTW Soroban. Ya can't skip the even numbers. They work too.

 

[soroban]

Hello, Ashley Cook!

First of all, I'll take a flying guess at what you meant . . .
$\displaystyle \;\;$(Edit: I see that Gene is on the same wavelength.)

What is the least positive interger $\displaystyle n$ such that $\displaystyle \,3^n\,-\,n^3$ is an even 4-digit number.

Second, what techniques are available to you?
$\displaystyle \;\;$Graphing? $\displaystyle \;$modulo artihmetic? $\displaystyle \;$ trial-and-error?

I noted that $\displaystyle 3^n$ is an odd number$\displaystyle \;\;\Rightarrow\;\;n^3$ must be odd$\displaystyle \;\;\Rightarrow\;\;n$ is odd.

Then try: $\displaystyle \,n\:=\:1,\,3,\,5,\,7,\,\cdots$

 

[Gene]

Yup, same wavelength all the way thru, but from the equation n³-3n=1000, n=10 is very close to the thousand so 11 is the first number to try and the analysis says it will work.