if a, b, and c are positive integers and ((19^a)^b)^c=19^19, what is the average of a, b, and c?

Question

if a, b, and c are positive integers and ((19^a)^b)^c=19¹⁹, what is the average of a, b, and c?

Answer 1

According to the assumption that [(19^a)^b]^c = 19^19, we can rewrite it as,

19^(abc) = 19^19

which can further be simplified to abc = 19.

Now consider that a,b and c are all positive integers, and 19 is a prime intrinsically, so the only option is that,

a*b*c = 1*1*19, i.e., a,b, and c can only equal 1, 1, and 19, respectively and in any order.

Therefore, the average of a,b and c should be (1+1+19)/3 = 7.