Determine all possible value(s) of a positive integer N that satisfies the following conditions:

(i) N is the product of three prime numbers, whose squares sum to 2331.

(ii) The sum of the positive divisors of N ( including 1 and N) is 10560.

The first criterion is satisfied by the following:

--primes--      N
11  19  43      8987
11  23  41      10373
11  29  37      11803
17  19  41      13243
23  29  31      20677

Of these, only the first also satisfies the second criterion.

So the only answer is N = 8987.

   10   for N1=1 to 15
   20   for N2=N1+1 to 15
   30   for N3=N2+1 to 15
   40    P1=prm(N1):P2=prm(N2):P3=prm(N3)
   50    Ts=P1*P1+P2*P2+P3*P3
   60    if Ts=2331 then
   70       :print P1;P2;P3,P1*P2*P3
   80       :Sd=1+P1+P2+P3+P1*P2+P1*P3+P2*P3+P1*P2*P3
   90       :if Sd=10560 then print "   ";P1;P2;P3,P1*P2*P3
  200   next
  210   next
  220   next
  

BTW, within the range checked (highest prime is 47, so its square doesn't exceed 2331), the only other N to satisfy the second criterion is 8729 ( = 7 * 29 * 43).  And a complete list of numbers that satisfy the second criterion is:

2  7   439   6146
2  31  109   6758
2  43  79    6794
3  5   439   6585
3  19  131   7467
3  23  109   7521
3  43  59    7611
7  11  109   8393
7  29  43    8729
11 19  43    8987

Posted by Charlie on 2009-10-18 15:38:15