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Digit Product Puzzle (Posted on 2015-09-03) Difficulty: 2 of 5
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?

No Solution Yet Submitted by K Sengupta    
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analytic solution | Comment 1 of 3
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

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