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1-1+1-1..

 
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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
 
Arjun Shastry
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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.
 
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