(c) (17)200=(18–1)200 We know that (x+a)n =xn+nxn–1.a +
n(n–1)
1×2
xn–2a2+
n(n–1)(n–2)
1×2×3
xn–3a3+.....+an 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)200is 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(∵200iseven) 1 is not divisible by 18 and is also less than 18. ∴1 is the remainder.