<PRE><B>Question 538527</B>
There may be a faster way.&lt;P&gt;
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.&lt;P&gt;
5^1 / 27 = remainder 5&lt;P&gt;
5^2:   /27 remainder 25&lt;P&gt;
5^3:  /27 = remainder 17&lt;P&gt;
5^4:  /27 = remainder 4&lt;P&gt;
5^5:  /27 = remainder 20&lt;P&gt;
5^6:  /27 = remainder 19&lt;P&gt;
5^7:  /27 = remainder 14&lt;P&gt;
5^8:  /27 = remainder 16&lt;P&gt;
5^9:  /27 = remainder 26&lt;P&gt;
5^10:  /27 = remainder 22&lt;P&gt;
5^11:  /27 = remainder 2&lt;P&gt;
5^12:  /27 = remainder 10&lt;P&gt;
5^13:  /27 = remainder 23&lt;P&gt;
5^14:  /27 = remainder 7&lt;P&gt;
5^15:  /27 = remainder 8&lt;P&gt;
5^16:  /27 = remainder 13&lt;P&gt;
5^17:  /27= remainder 11&lt;P&gt;
5^18:  /27= remainder 1&lt;P&gt;
5^19:  /27= remainder 5&lt;P&gt;
Finally!  The cycle is 18 powers of 5 up to 5^18.&lt;P&gt;
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.&lt;P&gt;
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&lt;P&gt;
For example 5^16.&lt;P&gt;
16/18 = 0 remainder 16.  So the remainder of (5^16)/27=13.&lt;P&gt;
5^24&lt;P&gt;
24/18 = 1 remainder 6.  So the remainder for (5^24)/27=19.&lt;P&gt;
For {{{5^(5^555)}}} it&#39;s (5^555)/18 which has a remainder of 17.  So the remainder of {{{(5^(5^555))/27}}} is 11.&lt;P&gt;
If you have a calculator (try wolframalpha.com) that can find the remainder of {{{(5^555)/18}}} then that calculator can find the remainder of {{{(5^(5^555))/27}}}.  For that, on that wolfram site, you&#39;d write 5^(5^555) mod 27 and it would spit out the answer.&lt;P&gt;
What if you don&#39;t have such a calculator?  Perform the same process as above to find the remainder pattern of {{{(5^555)/18}}}, use that to find the remainder of {{{(5^555)/18}}} and then use that to find the answer.&lt;P&gt;
5^1 / 18 = remainder 5&lt;P&gt;
5^2:   /18 remainder 7&lt;P&gt;
5^3:  /18 = remainder 17&lt;P&gt;
5^4:  /18 = remainder 13&lt;P&gt;
5^5:  /18 = remainder 11&lt;P&gt;
5^6:  /18 = remainder 1&lt;P&gt;
5^7:  /18 = remainder 5&lt;P&gt;
That one&#39;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&#39;ll use.&lt;P&gt;
R=1: 5
R=2: 7
R=3: 17
R=4: 13
R=5: 11
R=0: 1&lt;P&gt;
{{{(555)/6}}} has a remainder of 3.  Therefore the remainder of {{{(5^555)/18}}} is 17.  Plug that back into the first table (the 27&#39;s remainder table).&lt;P&gt;

That shows the remainder of {{{(5^(5^555))/27}}} is the R=17 in that table, which is 11.

&lt;P&gt;&lt;B&gt;If you need help understanding math so you can solve these problems yourself, then one on one online tutoring is the answer ($30/hr).  If you need faster solutions with guaranteed detailed answers, then go with personal problem solving ($3.50-$5.50 per problem).  Contact me at fcabanski@hotmail.com&lt;/B&gt;&lt;P&gt;</PRE>