Question 387076
Find the units digit for 7^197 without using a calculator.
:
Without using a calc you can find that the last digit is repeated every 4 power raises
7^1 = 7
7^2 = 9; last digit when you multiply previous last digit 7*7 = 9
7^3 = 3; last digit when you multiply previous last digit 7*9 = 3
7^4 = 1; last digit when you multiply previous last digit 7*3 = 1
7^5 = 7;  "    "   "  "  "  " "  7*1 = 7
7^6 = 9;  "    "   "  "  "  " "  7*7 = 9
7^7 = 3;  "    "   "  "  "  " "  7*9 = 3
7^8 = 1;  "    "   "  "  "  " "  7*3 = 1
7^9 = 7;  "    "   "  "  "  " "  7*1 = 7
7^10= 9;  "    "   "  "  "  " "  7*7 = 9
:
Can you see the pattern here? repeats after every 4th power
:
Using long diviion, find the remainder 197/4; that would be 1
Therefore the last digit should be 7