Test of Mathematics Solution Subjective 36 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
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Let be n numbers such that each is either 1 or -1. If then prove that 4 divides n.
Let us denote each product of four numbers as . For example . Note that the last one begins with indicating there are n ‘s.
We check the equation modulo 4 (that is in each step we change ‘something’ in the equation and check what happens to the remainder of the sum when divided by 4).In the first step we note that equation is 0 mod 4 (since 0 when divided by 4 gives 0 as the remainder).
Next we convert one of the a’s which is -1 into +1. Note that each appears in exactly 4 . Hence converting the one from -1 to +1 will alter the values of 4 Thus we may have five cases:
Thus we see that changing one -1 to +1 does not change the value of the total sum modulo 4. Similarly we change all negative one’s into positive one’s (and in each step the sum remains invariant in modulo 4) to get n. Thus . This implies 4 divides n.
Modular arithmetic, invariance principle.