Parity Condition on Sequence of 1 and -1

CAT 2000 Slot 1 · QA · Hard · Algebra

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Each of the numbers x1,x2,…,xnx_1, x_2, \dots, x_n (n≥4n \ge 4) is equal to 11 or −1-1. Suppose x1x2x3x4+x2x3x4x5+⋯+xnx1x2x3=0x_1 x_2 x_3 x_4 + x_2 x_3 x_4 x_5 + \dots + x_n x_1 x_2 x_3 = 0 then:

  1. A.

    nn is even

  2. B.

    nn is odd

  3. C.

    nn is an odd multiple of 3

  4. D.

    nn is prime

Answer

A

Explanation

There are nn terms in the sum, each being +1+1 or −1-1. For the total sum to be 00, there must be an equal number of +1+1s and −1-1s. Therefore, nn must be even (and in fact a multiple of 4).

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