In how many ways can 5^13 be expressed as a product of three natural numbers?

Also repeat the problem for 5^12.

There are 21 sets of three natural numbers that can express the product 5¹³:

{1, 1, 1220703125}, {1, 5, 244140625}, {1, 25, 48828125},
{5, 5, 48828125}, {1, 125, 9765625}, {5, 25, 9765625},
{1, 625, 1953125}, {5, 125, 1953125}, {25, 25, 1953125},
{1, 3125, 390625}, {5, 625, 390625}, {25, 125, 390625},
{1, 15625, 78125}, {5, 3125, 78125}, {25, 625, 78125},
{125, 125, 78125}, {5, 15625, 15625}, {25, 3125, 15625},
{125, 625, 15625}, {125, 3125, 3125}, {625, 625, 3125}

If order counts, there are 105 ways. E.g., the set
{1, 1, 1220703125} can be used to express 5¹³ three ways: 
   1 x 1 x 1220703125
   1 x 1220703125 x 1 
   1220703125 x 1 x 1. 

There are 19 sets of three natural numbers that can express the product 5¹³:

{1, 1, 244140625}, {1, 5, 48828125}, {1, 25, 9765625},
(5, 5, 9765625}, {1, 125, 1953125}, {5, 25, 1953125},
{1, 625, 390625}, {5, 125, 390625}, {25, 25, 390625},
{1, 3125, 78125}, {5, 625, 78125}, {25, 125, 78125},
{1, 15625, 15625}, {5, 3125, 15625}, {25, 625, 15625},
{125, 125, 15625}, {25, 3125, 3125}, {125, 625, 3125},
{625,625,625}

If order counts, there are 91 ways.

Posted by Dej Mar on 2009-11-23 02:23:37