SOLUTION: Find the remainder when 40^{13} is divided by 81.

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Question 1207676: Find the remainder when 40^{13} is divided by 81.
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
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40%5E13+%2F+81+=8285044938271604938+%2B22%2F81
remainder is 22

Answer by ikleyn(52772) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the remainder when  40%5E13  is divided by  81.
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        It is clear that the mathematical meaning of this problem is not to follow literally
        the written formula.

        Its meaning is to decrease / (to reduce)  the degrees and values of participating numbers
        to make calculations easier using standard properties of operations of modular arithmetic. 


Following this idea, I write  40 = 36 + 4,

    40%5E13 = %2836%2B4%29%5E13.


Next we should apply the Newtonian binomial formula.


It will give the sum of the terms  C%5B13%5D%5Ek%2A36%5Ek%2A4%5E%2813-k%29,  k = 0, 1, 2, 3, . . . , 13.


All the terms with k >= 2  will be zero by modulo 81, since 36 = 9*4.


Therefore, we can exclude all these terms from our consideration.


So, the terms under our consideration are the terms with k= 0 and k= 1, or

    C%5B13%5D%5E0%2A4%5E13 + C%5B13%5D%5E1%2A36%2A4%5E12 = 4%5E13 + 13%2A36%2A4%5E12.


This expression is easy to calculate using a regular calculator or Excel spreadsheet; 
its value is 7918845952.


Finally,  7918845952 mod 81 is 22  (use long division or Excel function mod)


So, the  ANSWER  is 22.

Solved.


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