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Math Help - Finding a minimum value for system of equations

  1. #1
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    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.
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  2. #2
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    Re: Finding a minimum value for system of equations

    Hello, crossingdouble!

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

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

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

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

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

    The least value of m occurs when k = 1.

    . . Hence: . m= 45

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

    Therefore: . m + n \:=\:45 + 15 \:=\:60 . . . answer (d)
    Thanks from crossingdouble
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  3. #3
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    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?
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