# Finding a minimum value for system of equations

• Jan 9th 2013, 06:31 PM
crossingdouble
Finding a minimum value for system of equations
I am stumped on this one.

Assume m and n are positive integers. 75m = n^3. What is the minimum possible value of m + n?
a) 15
b) 30
c) 50
d) 60
e) 5700

I was unsure of where to begin, so I tried to set up a system of equations and try each possible answer.
75m = n^3, m + n = 15
But, this gave me imaginary numbers when I tried solving each possible answer. I'm guessing that I'm going at this completely wrong.

I know that when you have a quadratic you can find the minimum by -b/2a when in the form y = ax^2 + bx + c, but I didn't think I could do this when I have a cube there.
• Jan 9th 2013, 08:27 PM
Soroban
Re: Finding a minimum value for system of equations
Hello, crossingdouble!

Quote:

$\displaystyle m\text{ and }n\text{ are positive integers. }\;75m \,=\, n^3$

$\displaystyle \text{What is the minimum possible value of }m + n\,?$

$\displaystyle (a)\;15 \qquad (b)\;30 \qquad (c)\;50\qquad (d)\;60 \qquad (e)\; 5700$

If $\displaystyle 75m$ is a cube, we have: .$\displaystyle 3\!\cdot\!5^2\!\cdot\! m \:=\:n^3$

Then $\displaystyle m$ must be of the form: .$\displaystyle m \,=\,3^2\!\cdot\!5\cdot\!k^3$
. . (Do you see why?)

The least value of $\displaystyle m$ occurs when $\displaystyle k = 1.$

. . Hence: .$\displaystyle m= 45$

And we have: .$\displaystyle 75\cdot45 \,=\,n^3 \quad\Rightarrow\quad n^3 \,=\,3375 \quad\Rightarrow\quad n \,=\,15$

Therefore: .$\displaystyle m + n \:=\:45 + 15 \:=\:60$ . . . answer (d)
• Jan 11th 2013, 06:20 PM
crossingdouble
Re: Finding a minimum value for system of equations
Hi, Soroban!

Thanks for your reply. Is the reason m must be in that form because the 3's and 5's must be in groups of 3?

So, if the problem said 225m = n^3, then m would be in the form m = 3 * 5 * k^3?

Oh, and I did work it out with a system of equations, and it checked out. I guess I was just doing those wrong?