Question 682117

7^12 - 4^12 


(7^6)^2 - (4^6)^2 


(7^6 - 4^6)(7^6 + 4^6) ... Use the difference of squares rule


((7^3)^2 - (4^3)^2)(7^6 + 4^6)


(7^3 - 4^3)(7^3 + 4^3)(7^6 + 4^6) ... Use the difference of squares rule


(7 - 4)(7^2 + 7*4 + 4^2)(7^3 + 4^3)(7^6 + 4^6) ... Use the difference of cubes rule


(7 - 4)(49 + 28 + 16)(7^3 + 4^3)(7^6 + 4^6)


(3)(93)(7^3 + 4^3)(7^6 + 4^6)


(3)(93)(7 + 4)(7^2 - 7*4 + 4^2)(7^6 + 4^6) ... Use the sum of cubes rule


(3)(93)(7 + 4)(49 - 28 + 16)(7^6 + 4^6)


(3)(93)(11)(37)(7^6 + 4^6)


(3*11)(93)(37)(7^6 + 4^6)


(33)(93)(37)(7^6 + 4^6)


We can see that 33 is a factor of (33)(93)(37)(7^6 + 4^6), so 33 is also a factor of 7^12 - 4^12 since each line in the work shown above is equivalent to any other line (ie they represent the same number)


This means that 7^12 - 4^12 is exactly divisible by 33.


Therefore, the answer is A) 33.