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Two binary operations $\oplus$ and $^\ast$ are defined over the set $\{a, e, f, g, h\}$ as per the following tables

$\oplus$ a e f g h
a a e f g h
e e f g h a
f f g h a e
g g h a e f
h h a e f g
$^\ast$ a e f g h
a a a a a a
e a e f g h
f a f h e g
g a g e h f
h a h g f e

Thus, according to the first table $f \oplus g = a$, while according to the second table $g \ast h = f$, and so on. Also, let $f^2 = f\ast f, \: g^3 = g \ast g \ast g$, and so on.

What is the smallest positive integer n such that $g^n = e?$

  1. $4$
  2. $5$
  3. $2$
  4. $3$
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g$^{n}$ = e. 

If n = 2, 

g$^{2}$ = g * g = h

If n = 3,

g$^{3}$  = g * g * g = h * g = f.                          // here '*' follows associative property.

If n = 4,

g$^{4}$  = g * g * g * g = h * g * g = f * g = e.   

Ans -A. minimum value of n = 4. 

 

 

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