.What is the remainder when dividing 38^213 by 13?

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SUMMARY

The problem of finding the remainder when dividing 38213 by 13 can be effectively solved using Fermat's Little Theorem. According to the theorem, since 13 is a prime number, we can simplify the exponent 213 modulo 12 (which is 13-1). This results in 213 mod 12 = 9. Therefore, 38213 mod 13 is equivalent to 389 mod 13. The final calculation yields a remainder of 5.

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Homework Statement



Find the remainder when dividing 38^{213} by 13.

Homework Equations


Fermats little theorem: a^{p-1}\equiv 1 Mod(p)

The Attempt at a Solution


I tried proving this with fermats little theorem or using the more general Euler theorem but I am overlooking some manipulation. To my dismay 12 does not divide 213 and I am not seeing how to put the question in the right form. ANy help is greatly apreciarted.
Tal
 
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