Question 365979: what is the last digit in the example 3 raised to the power of 2009?? and what is the solution?? is there a shortcut method to find the answers of the exponent
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Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! yes there is.
3^0 = 1 last digit is 1
3^1 = 3 last digit is 3
3^2 = 9 last digit is 9
3^3 = 27 last digit is 7
3^4 = 81 last digit is 1
3^5 = 243 last digit is 3
3^6 = 729 last digit is 9
3^7 = 2187 last digit is 7
3^8 = 6561 last digit is 1
3^9 = 19683 last digit is 3
3^10 = 59049 last digit is 9
3^11 = 177147 last digit is 7
3^12 = 531441 last digit is 1
3^13 = 1594323 last digit is 3
3^14 = 4782969 last digit is 9
3^15 = 14378907 last digit is 7
etc.
the last digits repeat every 4 exponents.
The remainder of dividing the exponent by 4 gives us the equivalent base exponent used.
Let R(e/4) be the remainder of dividing the exponent by 4.1
Examples:
if the exponent is 4, then the equivalent base exponent is R(4/4)) = 0.
if the exponent is 5, then the equivalent base exponent is R(5/4)) = 1.
if the exponent is 6, then the equivalent base exponent is R(6/4)) = 2.
if the exponent is 7, then the equivalent base exponent is R(7/4)) = 3.
if the exponent is 8, then the equivalent base exponent is R(8/4)) = 0.
if the exponent is 9, then the equivalent base exponent is R(9/4)) = 1.
if the exponent is 10, then the equivalent base exponent is R(10/4)) = 2.
if the exponent is 11, then the equivalent base exponent is R(11/4)) = 3.
if the exponent is 12, then the equivalent base exponent is R(12/4)) = 0.
if the exponent is 13, then the equivalent base exponent is R(13/4)) = 1.
if the exponent is 14, then the equivalent base exponent is R(14/4)) = 2.
if the exponent is 15, then the equivalent base exponent is R(15/4)) = 3.
etc......
if the base exponent is 0, then the last digit is 1.
if the base exponent is 1, then the last digit is 3.
if the base exponent is 2, then the last digit = 9.
if the base exponent is 3, then the last digit is 7.
using this formula:
if the exponent is 2009, then the equivalent base exponent is R(2009/4)) = 1.
What you get is 2009/4 = 502.25
.25 * 4 = 1
the remainder is 1.
the last digit is 3.
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