cho P=1^2017+2^2017+...+3^2017. Q = 1+2+3+...+2017. Chứng minh P chia hết cho Q?

Note that for n is odd , aⁿ + bⁿ is divided by a + b.

1²º¹⁷ + 2²º¹⁷ + 3²º¹⁷ + ... + 2017²º¹⁷
= (1²º¹⁷ + 2017²º¹⁷) + (2²º¹⁷ + 2016²º¹⁷) + (3²º¹⁷ + 2015²º¹⁷) + ... + (1008²º¹⁷ + 1010²º¹⁷) + 1009²º¹⁷
= 2018(a1) + 2018(a2) + 2018(a3) + ... + 2018(a1008) + 1009²º¹⁷ (a's are integers) is divided by 1009.

On the other hand 1²º¹⁷ + 2²º¹⁷ + 3²º¹⁷ + ... + 2017²º¹⁷
= (1²º¹⁷ + 2016²º¹⁷) + (2²º¹⁷ + 2015²º¹⁷) + (3²º¹⁷ + 2014²º¹⁷) + ... + (1008²º¹⁷ + 1009²º¹⁷) + 2017²º¹⁷
= 2017(b1) + 2017(b2) + 2017(b3) + ... + 2017(b1008) + 2017²º¹⁷ (b's are integers) is divided by 2017.

But 1009 and 2017 are relatively prime ,
so 1²º¹⁷ + 2²º¹⁷ + 3²º¹⁷ + ... + 2017²º¹⁷ is divided by 1009 × 2017 = 1 + 2 + 3 + ... + 2017.

bus