Question

What is the remainder when 4^103 is divided by 17?

(a) 10

(b) 14

(c) 13

(d) 16

The question was posed to me in an interview.

The origin of the question is Binomial Theorem for Positive Integral Index topic in chapter Binomial Theorem of Mathematics – Class 11

Answer 1

Right option is (d) 16

To explain: 4^103 = 4 x 4^102

4^103 = 4 x (4^2)^51

4^103 = 4 x (16)^51

4^103 = 4 x (17 – 1)^51

4^103 = 4 x \(\Sigma_{r = 0}^{r = 51}\)(51Cr 17^24 – r (-1)^r

4^103 = 4 x [51C0 17^51 (-1)^0 + 51C1 17^51 (-1)^1 +….+ 51C50 17^1 (-1)^50 + 51C51 17^0 (-1)^51]

4^103 = 4 x 17 x k – 1

The remainder = 17 – 1

Remainder = 16.