Problem Solving

Vincen wrote:What is the greatest prime factor of $$8^{12}-2^{30}?$$ A. 2
B. 3
C. 5
D. 7
E. 11

To determine the greatest prime factor of 8^12 - 2^30, we must find the prime factorization of 8^12 - 2^30

Given: 8^12 - 2^30
Rewrite 8 as 2³ to get: (2³)^12 - 2^30
Apply Power of Power rule to get: 2^36 - 2^30
Factor out 2^30 to get: 2^30(2^6 - 1)
Evaluate: 2^30(64- 1)
Simplify: 2^30(63)
Factor 63 to get: (2^30)(3)(3)(7)

At this point, we can see that the greatest prime factor is 7

Answer: D

Cheers,
Brent

Vincen wrote:What is the greatest prime factor of $$8^{12}-2^{30}?$$ A. 2
B. 3
C. 5
D. 7
E. 11

We can simplify the given expression:

(2^3)^12 - 2^30

2^36 - 2^30

2^30(2^6 - 1) = 2^30(63) = 2^30 x 9 x 7 = 2^30 x 3^2 x 7

So the largest prime factor is 7.

Answer: D