Tom, Dick and Harry were searching for 3-digit triangular numbers (numbers of the form k*(k+1)/2) that are themselves each the product of three different triangular numbers greater than 1 (so 1*3*15 = 45 doesn't count, nor does 3*10*10 = 300 because of the duplicated 10).

Each of them found a different triangular number. One of the triangular factors is found only in Harry's solution. Another of the triangular factors is found only in Tom's solution.

What are the three triangular factors making up Dick's solution?

I constructed a list of triangle numbers in a spreadsheet and proceeded to divide the list successively by the triangle number which I had currently at to top of my list.

Eventually I determined that there were three triangle numbers, 378, 630 and 990 which were products each of 3 different triangle numbers:
378 = 3 *  6  * 21
630 = 3 * 10 * 21  and
990 = 3 *  6  * 55.

Since Harry has a factor which is unique to him [10], his number is 630.
Tom has the factor [55] which neither of the others possesses so his number is 990.
Dick's triangular factors are therefore 3, 6 and 21 for his number 378.

Posted by brianjn on 2007-04-21 21:16:13