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How to find the number that's evenly divisible by a set of numbers from 1 to n?

  1. Aug 17, 2011 #1
    How can you go about finding the smallest number (x) that is evenly divisible by a set of numbers from 1 to n?

    For example:

    If n = 3, x=6
    n = 4, x=12
    n=5, x=60
    n=6, x=60


    This is easy to do when n is a small number, however how would one go about finding the answer if n=20 for example?

    I have written a program that tries to brute force the answer, but I'm almost certain there's a much more elegant way to do it. Any advice?
     
  2. jcsd
  3. Aug 17, 2011 #2
    Here's the program I wrote by the way, incase anyone is interested:
    Code (Text):

    class Problem5 {
     static int[] theArray;
    static int multiplyer = 11 ;
     static boolean test= false;
    static int current;


     //Makes array
     void maker (int number){
       for(int i=number; i>0; i--){
         theArray[i-1]=i;
         
       }
      }  
       
      public static void main (String[] args){
     
       Problem5 instance1 = new Problem5();
       theArray = new int[multiplyer];
       instance1.maker(multiplyer);
       current = theArray[theArray.length-1];
       System.out.println (current);
       
       while (test == false){
         for(int i = theArray.length; i>0; i--){
         if (current % theArray[i-1]>0){
           current = current + theArray[multiplyer -1];
           i=0;
           test = false;
         } else{
           System.out.println(current+" is divisible by "+ theArray[i-1]);
           test = true;
         }
        }  
       }
         
      }
         
    }
     
    Last edited by a moderator: Aug 17, 2011
  4. Aug 17, 2011 #3
    Phrased another way, your problem is to find the lowest common multiple (LCM) of a set of numbers from 1,...,n . Luckily, there is an easy formula to get the LCM for two numbers a and b from the greatest common divisor (GCD): LCM(a,b) = |a*b|/GCD(a,b) . I say luckily because finding the GCD is easy - just use the Euclidean algorithm. There's probably a snazzy way to deal with this for more than two numbers but you could adopt it by finding LCM(a,b), then LCM(LCM(a,b),c), etc. Check out http://en.wikipedia.org/wiki/Least_common_multiple

    If you stick with a brute force approach, remember that apart from {1}, each set contains 2. Think about what this means for the LCM...
     
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