Find the cycle of remainders. Find the remainder of 5^1. Multiply that remainder by 5. Divide by 27. Find that remainder. That will be the remainder of 5^2. Continue until there is a pattern. Use the pattern to find the given remainder.

5^1 / 27 = remainder 5

5^2: /27 remainder 25

5^3: /27 = remainder 17

5^4: /27 = remainder 4

5^5: /27 = remainder 20

5^6: /27 = remainder 19

5^7: /27 = remainder 14

5^8: /27 = remainder 16

5^9: /27 = remainder 26

5^10: /27 = remainder 22

5^11: /27 = remainder 2

5^12: /27 = remainder 10

5^13: /27 = remainder 23

5^14: /27 = remainder 7

5^15: /27 = remainder 8

5^16: /27 = remainder 13

5^17: /27= remainder 11

5^18: /27= remainder 1

5^19: /27= remainder 5

Finally! The cycle is 18 powers of 5 up to 5^18.

Divide the power, x, by 18. The remainder shows which number in the cycle to use for 5^x/27. The following table shows the corresponding remainder of (5^x)/27 for each remainder of x/18.

R=1: 5
R=2: 25
R=3: 17
R=4: 4
R=5: 20
R=6: 19
R=7: 14
R=8: 16
R=9: 26
R=10: 22
R=11: 2
R=12: 10
R=13: 23
R=14: 7
R=15: 8
R=16: 13
R=17: 11
R=0: 1

For example 5^16.

16/18 = 0 remainder 16. So the remainder of (5^16)/27=13.

5^24

24/18 = 1 remainder 6. So the remainder for (5^24)/27=19.

For it's (5^555)/18 which has a remainder of 17. So the remainder of is 11.

If you have a calculator (try wolframalpha.com) that can find the remainder of then that calculator can find the remainder of . For that, on that wolfram site, you'd write 5^(5^555) mod 27 and it would spit out the answer.

What if you don't have such a calculator? Perform the same process as above to find the remainder pattern of , use that to find the remainder of and then use that to find the answer.

5^1 / 18 = remainder 5

5^2: /18 remainder 7

5^3: /18 = remainder 17

5^4: /18 = remainder 13

5^5: /18 = remainder 11

5^6: /18 = remainder 1

5^7: /18 = remainder 5

That one's a lot shorter. The cycle of remainders for 5 to some power divided by 18 is 6 long. Divide the exponent by 6, find the remainder (you can find exponent modulo 6 in many calculators) and that remainder will tell you which of these 6 you'll use.

R=1: 5
R=2: 7
R=3: 17
R=4: 13
R=5: 11
R=0: 1

has a remainder of 3. Therefore the remainder of is 17. Plug that back into the first table (the 27's remainder table).

That shows the remainder of is the R=17 in that table, which is 11.

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