No answers exist for a + b + c = 43 and a^3 + b^3 + c^3 = 17299^2, even allowing them all to be negative: http://ideone.com/FTm8l
On Sat, Jul 16, 2011 at 6:28 PM, Leopoldo Taravilse <[email protected]> wrote: > You can solve it with no programming. > a+b+c = 43 && a^3 + b^3 + c^3 = 17299 > As 17299 == 3 mod 4 then as a^3,b^3 and c^3 can be 0 or 1 mod 4 then a, b > and c are odd. > 27^3 > 17299 so a < b < c <= 25 > 1^3 == 1 mod 5 > 2^3 == 3 mod 5 > 3^3 == 2 mod 5 > 4^3 == 4 mod 5 > 5^3 == 5 mod 5 (or 0^3 == 0 mod 5) > So as 17299 == 43 + 1 mod 5 at least one of a, b and c must be 2 mod 5 so > there are two options, 7 or 17. > Case 1 (7): a+b = 36 && a^3 + b^3 = 16956 > Case 2 (17) a+b = 26 && a^3 + b^3 = 12386 > Let's see Case 1 first: a and b can be 1, 3, 5, 9, 11, 13, 15, 17, 19, 21, > 23 or 25, but if a = 1, 3, 5 or 9 then b > 25 so a and b can be 11, 13, 15, > 17, 19, 21, 23 or 25. We can easily discard 13 and 23 because of modulo 5 > reasons so we have 11, 15, 17, 19, 21 or 25 so we can try with this six > numbers. > Case 2: a and b can be 1, 3, 5, 9, 11, 13, 15, 17, 19, 21 or 23, but we can > discard 3, 13 and 23 again so we have 1, 5, 9, 11, 15, 17, 19 and 21 to try. > > On Sat, Jul 16, 2011 at 6:54 AM, micke <[email protected]> wrote: >> >> What are the three positive integers whose sum is 43. and the sum of >> the cubes of three integers is square of the number 17299. >> >> -- >> You received this message because you are subscribed to the Google Groups >> "google-codejam" group. >> To post to this group, send email to [email protected]. >> To unsubscribe from this group, send email to >> [email protected]. >> For more options, visit this group at >> http://groups.google.com/group/google-code?hl=en. >> > > -- > You received this message because you are subscribed to the Google Groups > "google-codejam" group. > To post to this group, send email to [email protected]. > To unsubscribe from this group, send email to > [email protected]. > For more options, visit this group at > http://groups.google.com/group/google-code?hl=en. > -- You received this message because you are subscribed to the Google Groups "google-codejam" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/google-code?hl=en.
