Question 1137583
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            I have much more simple solution.



<pre>
First, notice that 255 = 256 - 1 = {{{4^4-1}}},  so  {{{4^4}}} = 255 + 1.


Second, notice that  {{{4^503}}} = {{{4^500*4^3}}} = {{{(4^4)^125*64}}}.


Therefore,  {{{4^503}}} = {{{(255+1)^125*64}}}.


Apply the Newton's binomial formula to present  {{{(255+1)^125}}} as the sum of degrees of the number 255 with integer coefficients.

All the terms of this binomial expansion will have the number 255 in positive degrees, except the last term, which is 1.


So, all the terms of this binomial expansion are divisible by 255, except the last term 1.


It means that when you multiply this expansion by 64, all the terms of this new expansion are divisible by 255, 
except the last term, which is 64.


It proves that the reminder of  {{{4^503}}} is 64, when it is divided by 255.
</pre>

Solved.


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By the way, it is a STANDARD method for solving such problems.


It works smoothly in many other similar problems.