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document.write( "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.
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document.write( "5^1 / 27 = remainder 5
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document.write( "5^2: /27 remainder 25
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document.write( "5^3: /27 = remainder 17
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document.write( "5^4: /27 = remainder 4
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document.write( "5^5: /27 = remainder 20
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document.write( "5^6: /27 = remainder 19
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document.write( "5^7: /27 = remainder 14
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document.write( "5^8: /27 = remainder 16
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document.write( "5^9: /27 = remainder 26
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document.write( "5^10: /27 = remainder 22
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document.write( "5^11: /27 = remainder 2
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document.write( "5^12: /27 = remainder 10
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document.write( "5^13: /27 = remainder 23
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document.write( "5^14: /27 = remainder 7
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document.write( "5^15: /27 = remainder 8
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document.write( "5^16: /27 = remainder 13
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document.write( "5^17: /27= remainder 11
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document.write( "5^18: /27= remainder 1
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document.write( "5^19: /27= remainder 5
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document.write( "Finally! The cycle is 18 powers of 5 up to 5^18.
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document.write( "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.
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document.write( "R=1: 5
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document.write( "R=2: 25
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document.write( "R=3: 17
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document.write( "R=4: 4
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document.write( "R=5: 20
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document.write( "R=6: 19
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document.write( "R=7: 14
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document.write( "R=8: 16
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document.write( "R=9: 26
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document.write( "R=10: 22
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document.write( "R=11: 2
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document.write( "R=12: 10
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document.write( "R=13: 23
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document.write( "R=14: 7
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document.write( "R=15: 8
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document.write( "R=16: 13
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document.write( "R=17: 11
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document.write( "R=0: 1
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document.write( "For example 5^16.
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document.write( "16/18 = 0 remainder 16. So the remainder of (5^16)/27=13.
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document.write( "5^24
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document.write( "24/18 = 1 remainder 6. So the remainder for (5^24)/27=19.
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document.write( "For it's (5^555)/18 which has a remainder of 17. So the remainder of
is 11.
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document.write( "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.
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document.write( "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.
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document.write( "5^1 / 18 = remainder 5
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document.write( "5^2: /18 remainder 7
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document.write( "5^3: /18 = remainder 17
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document.write( "5^4: /18 = remainder 13
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document.write( "5^5: /18 = remainder 11
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document.write( "5^6: /18 = remainder 1
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document.write( "5^7: /18 = remainder 5
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document.write( "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.
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document.write( "R=1: 5
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document.write( "R=2: 7
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document.write( "R=3: 17
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document.write( "R=4: 13
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document.write( "R=5: 11
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document.write( "R=0: 1
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document.write( " has a remainder of 3. Therefore the remainder of
is 17. Plug that back into the first table (the 27's remainder table).
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document.write( "That shows the remainder of is the R=17 in that table, which is 11.\r
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