B757 FLIGHT CREW REPORTS ENGINE FAILURE AT FL360 AND DIVERSION TO NEAREST SUITABLE ARPT.

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IN CRUISE AT FL360; 140 NM N OF ZZZ2 WHEN R ENG FAILED. BOTH MYSELF AND THE COPLT; WHO WAS THE PF FELT THE ROLLBACK. WE STARTED PUTTING IN TRIM; RUNNING CHKLISTS; NOTIFYING ATC AND DECLARING AN EMER. WE STARTED A DSCNT AND TURN-BACK TO DRY LAND. OUR DISPATCHER ADVISED THAT BOTH ZZZ2 AND ZZZ1 HAD GOOD WX. WE CHOSE ZZZ2 SINCE IT WAS CLOSER. I ELECTED TO NOT ATTEMPT A RESTART DUE TO THE CLOSE PROX OF THE ARPT AND THE UNKNOWN STATE OF THE ENG. I HAD THE RELIEF PLT REVIEW THE OVERWT LNDG INFO IN OUR QRH. ONCE HE DETERMINED THAT WE HAD PLENTY OF RWY; I ELECTED TO LAND IMMEDIATELY RATHER THAN BURN OFF FUEL. I DECIDED TO HAVE THE COPLT CONTINUE TO FLY AND LAND; SO THAT I COULD KEEP GOOD SITUATIONAL AWARENESS. THE LNDG WAS EXCELLENT AND CRASH/FIRE HAD NOTHING TO RPT. WE CONTINUED TAXIING TO THE RAMP. I FEEL OUR COMPANY'S RECENT TRAINING EMPHASIS ON ENG FAILURES ON THE TRACK SYS DEFINITELY HELPED US. OUR WORKING TOGETHER AS A TEAM; BOTH THE PLTS AND FLT ATTENDANTS HELPED US TO ACCOMPLISH EVERYTHING AND IN A TIMELY MANNER. SUPPLEMENTAL INFO FROM ACN 809509: AT 36000 FT ON THE TRACK SYS THE CAPT REQUESTED FOR ME TO GET BACK TO MY COCKPIT STATION; WHILE I WAS IN MY SCHEDULED CREW BREAK. WHEN I RETURNED TO THE COCKPIT I WAS INFORMED THAT THE R ENG HAD FAILED; WE HAD DECLARED AN EMER; AND WE WERE IN THE PROCESS OF DIVERTING. ALL THE PERTINENT CHKLIST WERE COMPLETED; FLT ATTENDANTS; AND PAX WERE BRIEFED; AND WE PROCEEDED TO LAND; SINGLE ENG; WITHOUT ANY FURTHER INCIDENT. WE FOUND OUT LATER THAT THERE WAS A MECHANICAL FAILURE IN THE ENG'S GEARBOX; WHICH CAUSED THE ENG FAILURE. I BELIEVE THIS EVENT WAS UNAVOIDABLE; AND I FEEL THAT THE ENTIRE CREW DID A FANTASTIC JOB COORDINATING OUR DIVERT. ATC WAS VERY ACCOMMODATING; AND DISPATCH DID WHAT THEY COULD OVER ACARS; TO HELP US OUT. I BELIEVE CRM WAS USED EXTREMELY WELL; AND THE WAY WE HANDLED THIS SITUATION IS A TEXT BOOK DEMO ON HOW TO HANDLE AN EMER.