RRB NTPC Mains or Stage 2 Model Paper 7 with solutions

(c) $(17)^{200} = (18 - 1)^{200}$We know that$(x + a)^n$$= x^n + n x^{n-1} a$$+ \frac{n(n-1)}{1 \times 2} x^{n-2} a^2 + \frac{n(n-1)(n-2)}{1 \times 2 \times 3} x^{n-3} a^3 + \dots + a^n$We see that all the terms on the R.H.S. except an has x as one of its factor and hence are divisible by x. So, (x + a)n is divisible by x or not will be decided by an.Let x = 18, a = –1 and n = 200∴$(18 - 1)^{200}$is divisible by 18 or not will depend$(-1)^{200}$ as all other terms in its expansion will be divisible by 18 because each of them will have 18 as one of their factors.$(-1)^{200} = 1 (\because 200\text{ is even})$ 1 is not divisible by 18 and is also less than 18.∴1 is the remainder.