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Sum of Reciprocals and Remainder

XAT 2014 · Quantitative Ability & Data Interpretation · Hard · Number System

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Read the following instruction carefully and answer the question that follows:

Expression ∑n=1131n\sum_{n=1}^{13} \frac{1}{n} can also be written as x13!\frac{x}{13!}. What would be the remainder if x is divided by 11?

  1. A.

    2

  2. B.

    4

  3. C.

    7

  4. D.

    9

  5. E.

    None of the above

Answer

D

Explanation

x=13!∑n=1131n=∑n=11313!nx = 13! \sum_{n=1}^{13} \frac{1}{n} = \sum_{n=1}^{13} \frac{13!}{n}. All terms in the sum contain factor 11 except 13!11=13×12×10!\frac{13!}{11} = 13 \times 12 \times 10!. x(mod11)≡(13×12×10!)(mod11)≡(2×1×(−1))(mod11)≡−2≡9x \pmod{11} \equiv (13 \times 12 \times 10!) \pmod{11} \equiv (2 \times 1 \times (-1)) \pmod{11} \equiv -2 \equiv 9.

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