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Question

The set {1,2,4,7,8,11,13,14} is a group under multiplication modulo 15. The inverses of 4 and 7 are respectively

A
3 and 13
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B
2 and 11
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C
4 and 13
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D
8 and 14
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Solution

The correct option is C 4 and 13
The set S={1,2,4,7,8,11,12,14} is a group under multiplication modulo 15.
The identity element for this group is e=1sinαxϵS, 1.x mod 15=x
Now let the inverse of 4 be 41
Now (4.41)mod 15 =e=1
Since (4.4) mod 15=1
41=4 (This inverse is unique)
Similarly let the inverse of 7 be 71
(7.71) mod 15=1
putting each element of set as 71 by trial and error we get
(7.13) mod 15 = 91 mod 15 =1
71=13
So, 41 and 71 are respectively 4 and 13.
Correct choice is (c)

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