7^0 = 01

7^1 = 07

7^2 = 49

7^3 = 343

7^4 = ..401

7^5 = ..807

7^6 = ..649

7^7 = ..543

7^8 = ..801

7^9 = ..607

7^10 = ..249

7^11 = ..743

7^12 = ..201

7^13 = ..407

7^14 = ..849

7^15 = ..943

7^16 = ..601

7^17 = ..207

7^18 = ..449

7^19 = ..143

7^20 = ..001

since 5775 = 20*288 +15 we have

7^5775 = ((7^20)288)*(7^15) has the same last digits as 7^15 that is the last three digits of 7^5775 are 943.