Divisibilty of $1^{101} + 2^{101} + 3^{101}+ 4^{101}+\cdots+2016^{101}$

Question

$1^{101} + 2^{101} + 3^{101}+ 4^{101}+\cdots+2016^{101}$ is divisible by which of the following?

$(A)$ $2014$

$(B)$ $2015$

$(C)$ $2016$

$(D)$ $2017$

Could someone share the approach to deal with such type of questions?

Answer 1

1) $$1^{101}+2016^{101}=(1+2016)A_1=2017A_1$$ $$2^{101}+2015^{101}=(2+2015)A_2=2017A_2$$ ... $$1008^{101}+1009^{101}=(1008+1009)A_{1008}=2017A_{1008}$$

2) $$1^{101}+2015^{101}=(1+2015)B_1=2016B_1$$ $$2^{101}+2014^{101}=(2+2014)B_2=2016B_2$$ ... $$2016^{101}=2016B$$ But $$2016 \not |1013^{101}$$ Similarly, for $2015$ and $2014$

Answer: $2017$