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Multiply Sum and Sum of Squares II (Posted on 2010-12-07) Difficulty: 3 of 5
Determine all possible values of a duodecimal (base 12) positive integer N less than 100,000 (base 12), such that N is obtained by multiplying the sum of its digits with the sum of squares of its digits.

Prove that these are the only possible values of N that exist.

No Solution Yet Submitted by K Sengupta    
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Solution computer search and analytic proof (spoiler) Comment 1 of 1

The numbers in base 12 are 1, 335 and 805.

 digits in base 12       decimal equivalent
0  0  0  0  0  0  1          1
0  0  0  0  3  3  5          473
0  0  0  0  8  0  5          1157

DEFDBL A-Z
CLS
DO
  n = n + 1
  FOR p = 0 TO 7
    d(p) = d(p) + 1
    IF d(p) < 12 THEN EXIT FOR
    d(p) = 0
  NEXT
  IF d(7) > 0 THEN EXIT DO
  sum = d(0) + d(1) + d(2) + d(3) + d(4) + d(5) + d(6)
  sumsq = d(0) * d(0) + d(1) * d(1) + d(2) * d(2) + d(3) * d(3) + d(4) * d(4) + d(5) * d(5) + d(6) * d(6)
  IF sum * sumsq = n THEN
    FOR p = 6 TO 0 STEP -1
      PRINT d(p);
    NEXT p
    PRINT TAB(30); n
  END IF
LOOP

For a base-12 number L digits long, the least the number can be is 12^(L-1). The sum of the digits is at most 11*L and the sum of the squares is at most 121*L, and the product of those two sums is at most 1331*L^2.

While 12^(L-1) increases exponentially, 1331*L^2 is only polynomial, so the former overtakes and exceeds the latter at a certain point:

number of digits      least possible number  greatest possible product of sums
                         (decimal)               (decimal)
 1                      1                      1331
 2                      12                     5324
 3                      144                    11979
 4                      1728                   21296
 5                      20736                  33275
 6                      248832                 47916
 7                      2985984                65219
 8                      35831808               85184
 9                      429981696              107811
 10                     5159780352             133100
 

So by the time you get to 6-digit base-12 numbers, the least possible value of the number, 248,832 in decimal, exceeds the greatest possible product of the stated sums, which is 47,916 in decimal.


  Posted by Charlie on 2010-12-07 15:18:32
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