SOLUTION: Find the least residue of 5^8 (mod 7). Show your steps.

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Question 1093280: Find the least residue of 5^8 (mod 7).
Show your steps.
Found 2 solutions by rothauserc, ikleyn:
Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
modulus example defining residue
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Given
a, there is only one value b between 0 and n−1 so that
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a ≡ b(mod n). We call b the residue of a modulo n, and write b = (a mod n).
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Note The expression a ≡ b(mod n) means that a-b is a multiple of n
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We are asked for the least residue of 5^8 (mod 7)
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5^8 / 7 = 390625 / 7 = 390621 with remainder of 4
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least residue of 5^8 (mod 7) is 4
:
Note 11 ≡ 4(mod 7) as well
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Answer by ikleyn(52754) About Me  (Show Source):
You can put this solution on YOUR website!
.
The remainders of dividing 5%5En by 7 form a periodic sequence

n             1  2  3  4  5  6  7  8
5%5En mod 7:    5, 4, 6, 2, 3, 1, 5, 4


with the period length of 6.


So if somebody somewhen will ask you to find the remainder of division 5%5E2017 by 7, you do not need to calculate 5%5E2017 to get the answer.

You simply will look into the periodical sequence of the period 6 and easily will find the required remainder.