Find the HCF of (3$$^{45}$$​– 1) and (3$$^{35}$$​ – 1).

Correct option is D

Given:

We need to find the HCF of the expressions:
​$$(3^{45} - 1) \text{ and } (3^{35} - 1)$$​​

Concept Used:

The HCF of expressions of the form $$(a^m - 1) \ and \ (a^n - 1)$$​ is given by:
​$$a^{\gcd(m, n)} - 1$$​​

where  $$\gcd(m, n)$$​is the greatest common divisor of m and n .

Solution:

Find $$\gcd(45, 35) :$$​​
The greatest common divisor of 45 and 35 is 5.

Substitute into the formula for HCF:
​$$HCF = 3^{\gcd(45, 35)} - 1$$​​

The HCF of $$(3^{45} - 1)$$​ and $$(3^{35} - 1) \ is \ 3^{5} - 1$$ = 243-1  = 242 .