Finding HCF of Powers: ($3^{45} - 1$) and ($3^{35} - 1$)

To find the Highest Common Factor (HCF) of numbers in the form of ($a^m - 1$) and ($a^n - 1$), we use the property:

HCF($a^m - 1$, $a^n - 1$) = $a^{\text{HCF}(m, n)} - 1$

Step 1: Identify Exponents

In this problem, we have:

• $a = 3$
• $m = 45$
• $n = 35$

Step 2: Calculate HCF of Exponents

We need to find the HCF of the exponents, 45 and 35.

• Factors of 45 are 1, 3, 5, 9, 15, 45.
• Factors of 35 are 1, 5, 7, 35.
• The Highest Common Factor (HCF) of 45 and 35 is 5.

So, $\text{HCF}(45, 35) = 5$.

Step 3: Apply the Property

Using the property mentioned earlier:

HCF($3^{45} - 1$, $3^{35} - 1$) = $3^{\text{HCF}(45, 35)} - 1$

Substitute the HCF of the exponents:

HCF = $3^5 - 1$

Step 4: Calculate the Final Result

Calculate $3^5$:

$3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$

Now, subtract 1:

HCF = $243 - 1 = 242$

Conclusion

The HCF of ($3^{45} - 1$) and ($3^{35} - 1$) is 242.