Question

Show that n⁴ − n³ + n² − n is divisible by 2 for all positive integers n.

Answer 1

n4−n3+n2−n=n3(n−1)+n(n−1)=n(n−1)(n2+1)

Note that n and n−1 are always  opposite s i.e. one of them is odd, and other is even ⟹n(n−1) is always divisible by 2⟹n(n−1)(n2+1) is always divisible by 2⟹n4−n3+n2−n is always divisible by 2

for eg :let n be =2 ,

2^4 - 2^3 +2 ^2 -2 = 16 -8 +4-2 = 8 + 2= 10 which is divisible by 2.

similarly all numbers (except 1 ) will be divisible