$$\text{ The remainder of } \frac{1^{2013}+2^{2013}+\cdots + 2012^{2013}+2013^{2013}}{2014} \text{ is ... }$$

Question

myininaya · Answer

So I only just began thinking about this question
And this question comes from the almighty @mayankdevnani

sparrow2 · Answer

2012 or 2013 in the degree?

mayankdevnani · Answer

2013

myininaya · Answer

$$\text{ The remainder of } \frac{1^{2013}+2^{2013}+\cdots + 2012^{2013}+2013^{2013}}{2014} \text{ is ... }$$

ganeshie8 · Answer

I'll just leave a small hint :

Pair up

myininaya · Answer

$$\frac{1^{2013}+2013^{2013}}{2014}+\frac{2^{2013}+2012^{2013}}{2014} +\frac{3^{2013}+2011^{2013}}{2014}+ \cdots + \ \frac{1006^{2013}+1008^{2013}}{2014} \ +\frac{1007^{2013}}{2014}$$

ganeshie8 · Answer

All the terms except the last cancel anfd if there is any justice at all, 1007^2013 must be easy to reduce

myininaya · Answer

$$\text{ how do you show } \frac{a^{2013}+(2014-a)^{2013}}{2014} \text{ is 0}$$

myininaya · Answer

I mean the remainder

sparrow2 · Answer

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