Question

What is the remainder when 2000^1000 is divided by 13?

Answer 1

Hey

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$2000 {}^{1000}$

Well, 2000 = 13k - 2 for some k

=> We can replace to get :

${2000}^{1000} \equiv {2}^{1000} \equiv {16}^{250} \equiv {3}^{250}$

$\equiv \: 3 \times {27}^{83} \equiv \: 3(mod \: 13)$

Hence, the remainder is : 3