Find the H.C.F of (5^225 - 1), (5^55 - 1)?(5^20 - 1)(5^25 - 1)(5^55 - 1)(5^5 - 1)(5^45 - 1)

To find the H.C.F (Highest Common Factor) of the expressions $5^{225} - 1$ and $5^{55} - 1$, we can use the following properties of powers:

Step 1: Break Down the Problem

We first note the general formula for finding the H.C.F of numbers of the form $a^m - 1$ and $a^n - 1$: $\text{H.C.F}(a^m - 1, a^n - 1) = a^{\text{H.C.F}(m,n)} - 1$ For our problem, we need to find: $\text{H.C.F}(5^{225} - 1, 5^{55} - 1)$

Step 2: Relevant Concepts

We need to compute $\text{H.C.F}(225, 55)$:

1. Finding H.C.F of 225 and 55:
   • The prime factorization of 225 is $15^2 = 3^2 \cdot 5^2$.
   • The prime factorization of 55 is $5 \cdot 11$.

Using the Euclidean algorithm:

1. $225 = 4 \times 55 + 5$
2. $55 = 11 \times 5 + 0$

This means $\text{H.C.F}(225, 55) = 5$.

Step 3: Analysis and Detail

Using the found H.C.F: $\text{H.C.F}(5^{225} - 1, 5^{55} - 1) = 5^{\text{H.C.F}(225, 55)} - 1 = 5^5 - 1$

Step 4: Verify and Summarize

Now we calculate $5^5 - 1$: $5^5 = 3125 \quad \Rightarrow \quad 5^5 - 1 = 3125 - 1 = 3124$

Final Answer

Thus, the H.C.F of $5^{225} - 1$ and $5^{55} - 1$ is: $\text{H.C.F}(5^{225} - 1, 5^{55} - 1) = 3124$