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

n             1  2  3  4  5  6  7  8
 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 by 7, you do not need to calculate to get the answer.

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



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