Question:
In the group \( G = \{ 1, 2, 3, 4, \times_5 \} \), the solution of \( 2^{-1} \times (3 \times 5^x) = 4 \) is
In the group \( G = \{ 1, 2, 3, 4, \times_5 \} \), the solution of \( 2^{-1} \times (3 \times 5^x) = 4 \) is
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When working in groups under modular arithmetic, always reduce terms modulo the group order and use inverses to solve equations.
Updated On: Apr 22, 2026
- 1
- 2
- 3
- None of these
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The Correct Option is A
Solution and Explanation
Step 1: Understand the operation in the group.
We are given the group \( G = \{1, 2, 3, 4, \times_5 \} \), which represents a group under multiplication modulo 5. This means the operations are performed modulo 5.
Step 2: Express the equation in terms of the group operation.
The equation given is: \[ 2^{-1} \times (3 \times 5^x) = 4 \] First, solve for \( 2^{-1} \), the multiplicative inverse of 2 modulo 5. By checking multiples of 2 modulo 5, we find: \[ 2 \times 3 = 6 \equiv 1 \, (\text{mod} \, 5) \] So, \( 2^{-1} = 3 \) modulo 5.
Step 3: Substitute the inverse into the equation.
Now substitute \( 2^{-1} = 3 \) into the equation: \[ 3 \times (3 \times 5^x) = 4 \] Simplifying: \[ 9 \times 5^x = 4 \]
Step 4: Reduce modulo 5.
Since we are working modulo 5, reduce the terms: \[ 9 \equiv 4 \, (\text{mod} \, 5) \] So the equation becomes: \[ 4 \times 5^x = 4 \]
Step 5: Solve for \( 5^x \).
Divide both sides by 4 (which is invertible modulo 5): \[ 5^x = 1 \] Since \( 5^x \equiv 1 \, (\text{mod} \, 5) \) for any \( x \), the solution is \( x = 1 \).
Step 6: Conclusion.
Thus, the solution to the equation is \( x = 1 \), corresponding to option (A).
We are given the group \( G = \{1, 2, 3, 4, \times_5 \} \), which represents a group under multiplication modulo 5. This means the operations are performed modulo 5.
Step 2: Express the equation in terms of the group operation.
The equation given is: \[ 2^{-1} \times (3 \times 5^x) = 4 \] First, solve for \( 2^{-1} \), the multiplicative inverse of 2 modulo 5. By checking multiples of 2 modulo 5, we find: \[ 2 \times 3 = 6 \equiv 1 \, (\text{mod} \, 5) \] So, \( 2^{-1} = 3 \) modulo 5.
Step 3: Substitute the inverse into the equation.
Now substitute \( 2^{-1} = 3 \) into the equation: \[ 3 \times (3 \times 5^x) = 4 \] Simplifying: \[ 9 \times 5^x = 4 \]
Step 4: Reduce modulo 5.
Since we are working modulo 5, reduce the terms: \[ 9 \equiv 4 \, (\text{mod} \, 5) \] So the equation becomes: \[ 4 \times 5^x = 4 \]
Step 5: Solve for \( 5^x \).
Divide both sides by 4 (which is invertible modulo 5): \[ 5^x = 1 \] Since \( 5^x \equiv 1 \, (\text{mod} \, 5) \) for any \( x \), the solution is \( x = 1 \).
Step 6: Conclusion.
Thus, the solution to the equation is \( x = 1 \), corresponding to option (A).
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