Which of the following is true ?

Options

A. (a) The number of common roots of `z^(144) = 1 and z^(24) = 1` is 24
B. (b) The number of common roots of `z^(360) = 1 and z^(315)= 1` is 45
C. (c) The number of roots common to `z^(24) = 1, z^(20) = 1` and `z^(56) = 1` is 4
D. (d) The number of roots common to `z^(27) = 1, z^(125) =1` and `z^(49) = 1` is 1

Explanation

To solve the problem of finding the number of common roots of the equations given, we will use the concept of the highest common factor (HCF) of the powers in the equations. Let's analyze each option step by step.

### Step-by-Step Solution:

1. **Option A: \( z^{144} = 1 \) and \( z^{24} = 1 \)**
To find the number of common roots, we calculate the HCF of 144 and 24.
- **Finding HCF(144, 24)**:
- Prime factorization of 144:
\( 144 = 2^4 \times 3^2 \)
- Prime factorization of 24:
\( 24 = 2^3 \times 3^1 \)
- HCF is obtained by taking the minimum power of each prime factor:
\( HCF = 2^{\min(4, 3)} \times 3^{\min(2, 1)} = 2^3 \times 3^1 = 24 \)
- **Conclusion**: The number of common roots is **24**.

2. **Option B: \( z^{360} = 1 \) and \( z^{315} = 1 \)**
We calculate the HCF of 360 and 315.
- **Finding HCF(360, 315)**:
- Prime factorization of 360:
\( 360 = 2^3 \times 3^2 \times 5^1 \)
- Prime factorization of 315:
\( 315 = 3^2 \times 5^1 \times 7^1 \)
- HCF is obtained by taking the minimum power of each prime factor:
\( HCF = 3^{\min(2, 2)} \times 5^{\min(1, 1)} = 3^2 \times 5^1 = 45 \)
- **Conclusion**: The number of common roots is **45**.

3. **Option C: \( z^{24} = 1 \), \( z^{20} = 1 \), and \( z^{56} = 1 \)**
We calculate the HCF of 24, 20, and 56.
- **Finding HCF(24, 20, 56)**:
- Prime factorization of 24:
\( 24 = 2^3 \times 3^1 \)
- Prime factorization of 20:
\( 20 = 2^2 \times 5^1 \)
- Prime factorization of 56:
\( 56 = 2^3 \times 7^1 \)
- HCF is obtained by taking the minimum power of each prime factor:
\( HCF = 2^{\min(3, 2, 3)} = 2^2 = 4 \)
- **Conclusion**: The number of common roots is **4**.

4. **Option D: \( z^{27} = 1 \), \( z^{125} = 1 \), and \( z^{49} = 1 \)**
We calculate the HCF of 27, 125, and 49.
- **Finding HCF(27, 125, 49)**:
- Prime factorization of 27:
\( 27 = 3^3 \)
- Prime factorization of 125:
\( 125 = 5^3 \)
- Prime factorization of 49:
\( 49 = 7^2 \)
- HCF is obtained by taking the minimum power of each prime factor:
Since there are no common prime factors,
\( HCF = 1 \)
- **Conclusion**: The number of common roots is **1**.

### Final Conclusion:
- All options are correct:
- Option A: 24
- Option B: 45
- Option C: 4
- Option D: 1