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 three primes that meet the first condition are:

• 11, 19, 43 --- 11²+19²+43² = 2331
• 11, 23, 41 --- 11²+23²+41² = 2331
• 11, 29, 37 --- 11²+29²+37² = 2331
• 17, 19, 41 --- 17²+19²+41² = 2331
• 23, 29, 31 --- 23²+29²+31² = 2331

Of these the only set of primes that meet the second condition is:

• 11, 19, 43 --- 1+11+19+43+(11*19)+(11*43)+(19*43)+(11*19*43) = 10560

11*19*43 = 8987; N is therefore 8987

Posted by Dej Mar on 2009-10-19 00:58:21