Find the HCF of `3^(100)-1and3^(120)-1`.

Answer

To find the HCF (Highest Common Factor) of the numbers \(3^{100} - 1\) and \(3^{120} - 1\), we can use the property of exponents and the difference of powers.

### Step-by-Step Solution:

1. **Identify the Numbers**:
We have two numbers:
\[ a = 3^{100} - 1 \]
\[ b = 3^{120} - 1 \]

2. **Use the Property of HCF**:
The HCF of \(a^m - 1\) and \(a^n - 1\) can be expressed as:
\[ \text{HCF}(a^m - 1, a^n - 1) = a^{\text{HCF}(m, n)} - 1 \]
Here, \(a = 3\), \(m = 100\), and \(n = 120\).

3. **Calculate HCF of the Exponents**:
We need to find \(\text{HCF}(100, 120)\):
- The prime factorization of 100 is \(2^2 \times 5^2\).
- The prime factorization of 120 is \(2^3 \times 3^1 \times 5^1\).
- The common factors are \(2^2\) and \(5^1\).
- Therefore, \(\text{HCF}(100, 120) = 20\).

4. **Apply the HCF Property**:
Now we can substitute back into our HCF formula:
\[ \text{HCF}(3^{100} - 1, 3^{120} - 1) = 3^{20} - 1 \]

5. **Final Result**:
Thus, the HCF of \(3^{100} - 1\) and \(3^{120} - 1\) is:
\[ \text{HCF} = 3^{20} - 1 \]