Obgu n naq o pnaabg or cevzr. Vs gurl jrer, Crgre jbhyq vzzrqvngryl xabj gur nafjre. Vs nal bar jnf n cevzr terngre guna 50, gura Crgre jbhyq or noyr gb fbyir vg nf jryy, hfvat gur snpg gung n naq o yvr orgjrra 2 naq 99.
Fnenu xabjf Crgre pnaabg havdhryl qrgrezvar n naq o. Fb, n + o pnaabg or gur fhz bs gjb cevzrf, abe pna vg or gur fhz bs bs n cevzr terngre guna 50 naq nabgure ahzore orgjrra 2 naq 99. Gur fznyyrfg cevzr terngre guna 50 vf 53, naq 53+2 = 55. Fb vs n + o >= 55, gura gurer jbhyq or n cbffvoyr cnve jubfr cebqhpg unf n havdhr snpgbevmngvba va bhe enatr. Guvf erznvaf gehr sbe n naq o > 97, fvapr nyy gubfr cbffvoyr cnvef unir qvfgvapg cebqhpgf.
Fb, gur cbffvoyr inyhrf bs n + o ner ahzoref sebz 4 gb 54, juvpu pnaabg or rkcerffrq nf gur fhz bs cevzrf. Guvf ryvzvangrf gur rira ahzoref nf jryy nf frireny bqqf. Vg fgvyy yrnirf frireny pnfrf gb rknzvar.
Fvapr Crgre xabjf gur ahzoref bapr Fnenu erirnyf ur pbhyqa’g unir qrqhprq vg sebz gur cebqhpg nybar, gung zrnaf gung bs nyy gur cbffvoyr jnlf bs snpgbevat vg vagb gjb ahzoref, bayl bar erfhygf va n cnve jubfr fhz vfa’g gur fhz bs cevzrf.
Fvapr Fnenu xabjf gur fbyhgvba nsgre Crgre naabhaprf ur sbhaq vg, gurer pna bayl or bar cnve nzbat gubfr gung fhz gb n + o jvgu gur nobir cebcregl, v.r. jurer nal qvssrerag snpgbevmngvba yrnqf gb gur fhz orvat gur fhz bs gjb cevzrf.
Sebz urer V cebprrqrq gb rknzvar gur qvssrerag pnfrf. V gbbx npprcgnoyr inyhrf sbe n + o naq purpxrq jurgure gur nobir pevgrevn jrer fngvfsvrq. Gurer znl or n zber ryrtnag nccebnpu gung shegure fvzcyvsvrf guvatf, ohg V pbhyqa’g guvax bs vg.
V oryvrir n = sbhe, o = guvegrra jbexf va guvf pnfr. Fvapr vg pna or snpgberq gjb jnlf, Crgre qbrfa’g xabj gur ahzoref. Fnenu xabjf ur qbrfa’g xabj, orpnhfr gurve fhz vf abg gur fhz bs gjb cevzrf. Gur bgure snpgbevat, (gjb, gjragl fvk), unf n fhz bs gjragl rvtug juvpu vf gur fhz bs gjb cevzrf, fb Fnenu’f fgngrzrag zrnaf gung ur pna ryvzvangr gung bar. Fb, Crgre xabjf jung vg vf. V ybbxrq ng bgure jnlf bs jevgvat friragrra nf gur fhz bs gjb vagrtref sebz gjb gb avargl avar, naq guvf jnf gur bayl bar gung ranoyrq Crgre gb havdhryl qrgrezvar uvf snpgbevat tvira Fnenu’f vasbezngvba.
Very well done, although it seems you guessed the answer and then proved that that answer worked. How did you come up with that solution?
V svtherq guhf:
Obgu n naq o pnaabg or cevzr. Vs gurl jrer, Crgre jbhyq vzzrqvngryl xabj gur nafjre. Vs nal bar jnf n cevzr terngre guna 50, gura Crgre jbhyq or noyr gb fbyir vg nf jryy, hfvat gur snpg gung n naq o yvr orgjrra 2 naq 99.
Fnenu xabjf Crgre pnaabg havdhryl qrgrezvar n naq o. Fb, n + o pnaabg or gur fhz bs gjb cevzrf, abe pna vg or gur fhz bs bs n cevzr terngre guna 50 naq nabgure ahzore orgjrra 2 naq 99. Gur fznyyrfg cevzr terngre guna 50 vf 53, naq 53+2 = 55. Fb vs n + o >= 55, gura gurer jbhyq or n cbffvoyr cnve jubfr cebqhpg unf n havdhr snpgbevmngvba va bhe enatr. Guvf erznvaf gehr sbe n naq o > 97, fvapr nyy gubfr cbffvoyr cnvef unir qvfgvapg cebqhpgf.
Fb, gur cbffvoyr inyhrf bs n + o ner ahzoref sebz 4 gb 54, juvpu pnaabg or rkcerffrq nf gur fhz bs cevzrf. Guvf ryvzvangrf gur rira ahzoref nf jryy nf frireny bqqf. Vg fgvyy yrnirf frireny pnfrf gb rknzvar.
Fvapr Crgre xabjf gur ahzoref bapr Fnenu erirnyf ur pbhyqa’g unir qrqhprq vg sebz gur cebqhpg nybar, gung zrnaf gung bs nyy gur cbffvoyr jnlf bs snpgbevat vg vagb gjb ahzoref, bayl bar erfhygf va n cnve jubfr fhz vfa’g gur fhz bs cevzrf.
Fvapr Fnenu xabjf gur fbyhgvba nsgre Crgre naabhaprf ur sbhaq vg, gurer pna bayl or bar cnve nzbat gubfr gung fhz gb n + o jvgu gur nobir cebcregl, v.r. jurer nal qvssrerag snpgbevmngvba yrnqf gb gur fhz orvat gur fhz bs gjb cevzrf.
Sebz urer V cebprrqrq gb rknzvar gur qvssrerag pnfrf. V gbbx npprcgnoyr inyhrf sbe n + o naq purpxrq jurgure gur nobir pevgrevn jrer fngvfsvrq. Gurer znl or n zber ryrtnag nccebnpu gung shegure fvzcyvsvrf guvatf, ohg V pbhyqa’g guvax bs vg.