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Cyclicity of Unit Digits

XAT 2018 · Quantitative Ability & Data Interpretation · Medium · Quantitative Aptitude

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X and Y are the digits at the unit's place of the numbers (408X) and (789Y) where X ≠ Y. However, the digits at the unit's place of the numbers (408X)63(408X)^{63} and (789Y)85(789Y)^{85} are the same. What will be the possible value(s) of (X + Y)?

  1. A.

    9

  2. B.

    10

  3. C.

    11

  4. D.

    12

  5. E.

    None of the above

Answer

B

Explanation

Exponents modulo 4: 63≡3(mod4)63 \equiv 3 \pmod 4, 85≡1(mod4)85 \equiv 1 \pmod 4. We need unit digit of X3=Y1=YX^3 = Y^1 = Y. Checking pairs (X,Y)(X, Y) where X≠YX \neq Y and X3≡Y(mod10)X^3 \equiv Y \pmod{10}:

  • X=2  ⟹  Y=8X = 2 \implies Y = 8 (sum = 10)
  • X=3  ⟹  Y=7X = 3 \implies Y = 7 (sum = 10)
  • X=7  ⟹  Y=3X = 7 \implies Y = 3 (sum = 10)
  • X=8  ⟹  Y=2X = 8 \implies Y = 2 (sum = 10) In all possible valid cases, X+Y=10X + Y = 10.

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