Answer: Number 73 soldiers survives at last

Explanation: 
1st soldier has the sword and kills 2nd soldier, passes it to 3rd 
Round 1 survivors  
Round 1: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99
Round 2 survivors 
Round 2: 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97
Round 3 survivors 
Round 3: 1, 9, 17, 25, 33, 41, 49, 57, 65, 73
Round 4 survivors 
Round 4: 9, 25, 41, 57, 73, 89
Round 5 survivors 
Round 5: 9, 41, 73
Round 6 survivors 
Round 6: 9, 73
Round 7 survivors 
Round 7: 73