Data Sufficiency

Hello,

Can you please tell me how we can solve for C:

If k & j are positive integers, and k is not equal to j, does k have more different prime factors
than j?

(1) k = (10)^j
(2) j = (30)^k

OA: C


I tried to solve as follows:

1) Let j = 1 => k = 10^1 = 10
Here j has no prime factors while k has 2 prime factors i.e. 2 and 5
Hence, answer to our original question is yes

Let j = 30 => k = 10^30
Here j has three prime factors i.e. 2, 3 and 5 while k has 2 prime factors i.e. 2 and 5
Hence, answer to our original question is no

Hence, In-suff.


2) Let k = 1 and j = 30
Here k has no prime factor while j has 3 prime factors i.e. 2, 3 and 5
Here, answer to our original question is no

Let k = 210 and j = 30
Here k has 4 prime factors while j has 3 prime factors
Hence, answer to our original question is yes

Hence, in-suff.


From 1 and 2:

k = 10^j and j = 30^k

I was wondering how we can prove that C is suff. since we now have

k = 10^j = 10^(30^k)

Can you please assist? Thanks a lot.


Best Regards,
Sri