N is a 6-digit positive integer whose product of the digits is 2520.
What is the probability that the sum of the digits of N is a perfect square?
The below is wrong; see follow-up.
The prime factors of 2520 are
2 2 2 3 3 5 7
To make a 6-digit number we need to combine (multiply) at least two of the digits. The 5 and the 7 need to remain untouched.
digits arrangements weight sum
7 5 9 8 1 1 360 3 31
7 5 9 4 2 1 720 6 28
7 5 9 2 2 2 120 1 27 *
7 5 6 6 2 1 360 3 27 *
7 5 6 3 4 1 720 6 26
7 5 6 3 2 2 360 3 25 *
7 5 3 3 8 1 360 3 27 *
7 5 3 3 4 2 360 3 24
---
28
The sum of the weights of the square-summed digit sets (starred) is 10. The answer is 10/28 = 5/14.
We might call the type of factoring done a KenKen factoring: all the ways of factoring into single-digit numbers the set of which has a given cardinality.
Edited on September 3, 2015, 4:31 pm
|
|
Posted by Charlie
on 2015-09-03 11:06:44 |
analytic solution
