Homework Help Forum: arithmetic

Posted by Loyd on Monday, October 5, 2009 at 4:09am.

Find the smallest positive integer P such that the cube root of 400 times P is an integer.

• arithmetic - jim, Monday, October 5, 2009 at 4:52am

  This sounds awkward, but is actually easy. Consider the number 400P. It must end in zero - actually, it must end in 00, since it has 100 as a factor. Therefore its cube root must end in zero. Check this for yourself: the last digit of a cube can be determined by the last digit of the cube root. 1x1x1 = 2, 2x2x2=8 and so on, regardless of the higher digits. So for the cube root we're already down to factors of 10 - we only have to look at 10, 20, 30... and their cubes all end in 000 because of the three 10s in the factors. Does 10 work? No, 1000 is a cube of 10, but 400 doesn't divide evenly. Now try 20.

  
  

• arithmetic - PsyDAG, Monday, October 5, 2009 at 10:09am

  20 * 20 = 400, so 20 is only the square root of 400, not the cube root. Unfortunately, I can only tell you that 20 is not the answer, but I do not know the answer. Sorry.

  
  

• arithmetic - Reiny, Monday, October 5, 2009 at 10:48am

  400 = 2x2x2 x 2 x 5x5 every triplet of "same factors" will produce a cube so to make the right side of the above statement into a perfect cube, we need another 2x2x5 or 20 So let's multiply both sides by 20 400x20 = 2x2x2 x 2x2x2 x5x5x5 400(20) = 8000 400P = 8000 so P = 20

  
  

• arithmetic - Dr Russ, Monday, October 5, 2009 at 10:55am

  I think the easiest is to say the cuberoot(400P) =cuberoot(2x2x2x2x5x5xP) =2cuberoot(2x5x5xP) for cuberoot(2x5x5xP) to be an integer then P=2x2x5=20 so why is 20 not the answer?

  
  

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