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The number of solutions of the equation x2 = 1 in the ring ℤ/105ℤ is

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CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)
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  1. 0
  2. 2
  3. 4
  4. 8

Answer (Detailed Solution Below)

Option 4 : 8
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Detailed Solution

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Concept:

If xn = 1 then o(x) divides n

Explanation:

ℤ/105ℤ ≅ ℤ105

105 = 3 × 5 × 7

So \(U_{ℤ_{105}}\) ≅ U(3) × U(5) × U(7) ≅ ℤ2 × ℤ4 × ℤ6

Given x2 = 1 so o(x) divides 2 Hence o(x) = 1 or 2

Element of ℤ2 of order 1 and 2 is 2

Element of ℤ4 of order 1 and 2 is 2

Element of ℤ6 of order 1 and 2 is 2

Hence total such elements = 2 × 2 × 2 = 8 

Option (4) is correct

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