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Remainder of Power Expression Modulo 7

XAT 2020 · Quantitative Ability & Data Interpretation · Medium · Quantitative Aptitude

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What is the remainder if 1920−201919^{20} - 20^{19} is divided by 7?

  1. A.

    5

  2. B.

    0

  3. C.

    3

  4. D.

    1

  5. E.

    6

Answer

A

Explanation

19≡5≡−2(mod7)19 \equiv 5 \equiv -2 \pmod 7. 1920≡(−2)20=220(mod7)19^{20} \equiv (-2)^{20} = 2^{20} \pmod 7. Since 23≡1(mod7)2^3 \equiv 1 \pmod 7, 220=(23)6⋅22≡16⋅4=4(mod7)2^{20} = (2^3)^6 \cdot 2^2 \equiv 1^6 \cdot 4 = 4 \pmod 7.

20≡6≡−1(mod7)20 \equiv 6 \equiv -1 \pmod 7. 2019≡(−1)19=−1≡6(mod7)20^{19} \equiv (-1)^{19} = -1 \equiv 6 \pmod 7.

1920−2019≡4−(−1)=5(mod7)19^{20} - 20^{19} \equiv 4 - (-1) = 5 \pmod 7. Remainder is 5.

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