Most Divisors

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Problem Description

Which positive 3-digit integer has the most positive integer divisors?

Source:  Based on "2001 Mensa Puzzle Calendar" puzzle for October 13

Background & Techniques

Here's a problem that's simple to solve with a program but not easy with pencil & paper.   There may be a trick to solve it simply by hand, but I haven't found it.  

For this program, we'll just try all integers, n,  from 100 to 999 in a loop and for each n check all divisors from 1 to  n / 2.   To satisfy the requirements of the problem, all we have to do is check the number of divisors found for each n against maxdivisors, the maximum found so far. and save  n and maxdivisors whenever a higher value is found.  

 Delphi's remainder function. mod, is the easy way to check for divisors, if "number mod trialdivisor = 0" then trialdivisor is a divisor.  

We'll add a few more lines of code here to display the divisors, just in case someone wants to check our answer. 

Running/Exploring the Program 

Suggestions for Further Explorations

What is the maximum number of divisors for 4 digit numbers? 
5 digit numbers?  (Be warned - using this technique for the 5 digits case will execute about a billion trial divisions, so be prepared to wait for a few of minutes.)
How could the algorithm be made faster?  Hint: each successful division should give us 2 divisors and let us lower the upper limit of potential divisors that we need to check.

 

  

 
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