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Thread: smallest value?

  1. #1
    Within my grasp! infinityzero's Avatar
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    smallest value?

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    n and p are integers greater than 1
    5n is the square of a number
    75np is the cube of a number.
    The smallest value for n + p is
    A. 14
    B. 18
    C. 20
    D. 30
    E. 50






    Ans:A
    I will win

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    5n= the square of a number
    it is not difficult to see n=5.

    75*n*p
    =75*5*p
    =3*5*5*5*p
    =3*p*5^3=the cube of a number.
    In order to get a cube, all the prime components (i.e. 3 and 5 here) have to be to the 3rd power. Therefore, the smallest number for p is 3^2 =9.

    Therefore, n+p= 5 +9 =14 (A).

    Quote Originally Posted by infinityzero
    n and p are integers greater than 1
    5n is the square of a number
    75np is the cube of a number.
    The smallest value for n + p is
    A. 14
    B. 18
    C. 20
    D. 30
    E. 50






    Ans:A

  3. #3
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    "5n= the square of a number
    it is not difficult to see n=5."

    How does n = 5???

    I did 5n = x^2 and 75np = x^3, where x is "a number". I divided 75np by 5n to get 15p, and 15p = x:

    (75np/5n) = (x^3/x^2) --> 15p = x

    Since x represents a number (and not an actual variable in this problem), I said that 15 = p/x or 15 = p.

    Then 75*(15)*n = 5*5*3*5*3*n
    p = 9 and I would have to solve for n.

    I don't understand how n=5 automatically here though. Can someone explain? Thanks.
    Last edited by infinityzero; 12-31-2005 at 02:06 AM. Reason: Automerged post

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    Ride the Lightning !! kronique's Avatar
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    Quote Originally Posted by infinity
    "5n= the square of a number
    it is not difficult to see n=5."

    How does n = 5??
    n is five here because we need to find the smallest vaule of n+p this is possible only when both n & p are min.
    obviously for 5n to be a square we can have many values on n...namely 5, 20,80 etc.
    but we are interested in the smallest value of n. Hence n=5.


    I did 5n = x^2 and 75np = x^3, where x is "a number". I divided 75np by 5n to get 15p, and 15p = x:

    (75np/5n) = (x^3/x^2) --> 15p = x
    The highlighted part is wrong infinity, because by that statement,you are assuming that 5n is the square of x while 75np is the cube of the same no. x !!
    (This need'nt be always the case.You are considering a very special case.)
    Thinking more generally, as cybernike pointed out n=5.
    Now we have to find out the smallest value of p.
    as cybernike says , again..

    75*n*p
    =75*5*p
    =3*5*5*5*p
    =3*p*5^3=the cube of a number.
    In order to get a cube, all the prime components (i.e. 3 and 5 here) have to be to the 3rd power. Therefore, the smallest number for p is 3^2 =9.
    now from this, only if p=9,73, etc can the no. 3*p*(5^3) be the cube of any possible no.

    Again the lowest allowed value of p as we see is 9. Hence n =5 & p=9. so the smallest value of n+p = 14.

    Hope this is clear.

  5. #5
    Within my grasp! infinityzero's Avatar
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    Yes, ok. I understand. I was writing things out in an equation format, hoping to solve for n or p. But it seems that is not the case for this problem and 5n and 75np must be taken by themselves alone and not combine together to solve for n or p, which is what I thought was to be done (and am familiar with).

    Well, thank you both
    I will win

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