PKN wrote:

\(X= 1^{51}+2^{51}+..............+16^{51}\)

What is the remainder when X is divided by 17?

(a) 0

(b) 3

(c) 10

(d) 13

(e) 16

The concepts that you need to use for this question are of

Remainder Theorem & of

negative Remainders.

\(\frac{16^{51}}{17}\) gives a remainder which is same as the remainder of \(\frac{(-1)^{51}}{17}\).

Lets denote it is R(\(\frac{16^{51}}{17}\)) = R(\(\frac{(-1)^{51}}{17}\)) = - R(\(\frac{(1)^{51}}{17}\))

Similarly, R(\(\frac{15^{51}}{17}\)) = R(\(\frac{(-2)^{51}}{17}\)) = - R(\(\frac{(2)^{51}}{17}\))

R(\(\frac{14^{51}}{17}\)) = R(\(\frac{(-3)^{51}}{17}\)) = - R(\(\frac{(3)^{51}}{17}\))

R(\(\frac{13^{51}}{17}\)) = R(\(\frac{(-4)^{51}}{17}\)) = - R(\(\frac{(4)^{51}}{17}\))

R(\(\frac{12^{51}}{17}\)) = R(\(\frac{(-5)^{51}}{17}\)) = - R(\(\frac{(5)^{51}}{17}\))

R(\(\frac{11^{51}}{17}\)) = R(\(\frac{(-6)^{51}}{17}\)) = - R(\(\frac{(6)^{51}}{17}\))

R(\(\frac{10^{51}}{17}\)) = R(\(\frac{(-7)^{51}}{17}\)) = - R(\(\frac{(7)^{51}}{17}\))

R(\(\frac{9^{51}}{17}\)) = R(\(\frac{(-8)^{51}}{17}\)) = - R(\(\frac{(8)^{51}}{17}\))

Now you can see that each of the remainders of the last 8 terms is going to be equal & opposite in sign to the remainders of the first 8 terms.

They will cancel out & the Answer will be 0.

Although you do not need to do all the above calculations, it is pretty clear after you check for the last two terms.

I will explain the next steps too, for more clarity

R(\(\frac{X}{17}\)) = R(\(\frac{(1)^{51}}{17}\)) + R(\(\frac{(2)^{51}}{17}\)) + R(\(\frac{(3)^{51}}{17}\)) + R(\(\frac{(4)^{51}}{17}\)) + R(\(\frac{(5)^{51}}{17}\)) + R(\(\frac{(6)^{51}}{17}\)) + R(\(\frac{(7)^{51}}{17}\)) + R(\(\frac{(8)^{51}}{17}\)) + R(\(\frac{(9)^{51}}{17}\)) + R(\(\frac{(10)^{51}}{17}\)) + R(\(\frac{(11)^{51}}{17}\)) + R(\(\frac{(12)^{51}}{17}\)) + R(\(\frac{(13)^{51}}{17}\)) + R(\(\frac{(14)^{51}}{17}\)) + R(\(\frac{(15)^{51}}{17}\)) + R(\(\frac{(16)^{51}}{17}\))

R(\(\frac{X}{17}\)) = R(\(\frac{(1)^{51}}{17}\)) + R(\(\frac{(2)^{51}}{17}\)) + R(\(\frac{(3)^{51}}{17}\)) + R(\(\frac{(4)^{51}}{17}\)) + R(\(\frac{(5)^{51}}{17}\)) + R(\(\frac{(6)^{51}}{17}\)) + R(\(\frac{(7)^{51}}{17}\)) + R(\(\frac{(8)^{51}}{17}\)) - R(\(\frac{(8)^{51}}{17}\)) - R(\(\frac{(7)^{51}}{17}\)) - R(\(\frac{(6)^{51}}{17}\)) - R(\(\frac{(5)^{51}}{17}\)) - R(\(\frac{(4)^{51}}{17}\)) - R(\(\frac{(3)^{51}}{17}\)) - R(\(\frac{(2)^{51}}{17}\)) - R(\(\frac{(1)^{51}}{17}\))

Hence, R(\(\frac{X}{17}\)) = 0

Answer A.

Thanks,

GyM