Each of a1,a2,a3..aN are either 1 or -1. The expression S = a1*a2*a3*a4+a2*a3*a4*a5+....+aN*a1*a2*a3 = 0 (* means multiplication and a1*a2 means value of a1 multiplied by value of a2) Prove that 4 divides N. I have taken this from Problem Solving Strategies
posted 12 years ago
If sign of every a(i) is changed,nothing changes .If sign of two a(i)s are same and other two are of different,nothing changes.If sign of one or three a(i) are different then S changes by +-4.If all four a(i) are of same sign,then S changes by +-8.That means S is changing by +-4 or +-8. S=0mod(4).But S = n hence 4 divides n.