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
_________________