No en los enteros, pero al parecer si en los reales: http://www.wolframalpha.com/input/?i=+a+%2B+b+%2B+c+%3D+43+and+a^3+%2B+b^3+%2B+c^3+%3D+17299^2
2011/7/16 Luke Pebody <[email protected]> > 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. > > -- 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.
