I have got one inductive proof of the F.T.A., as in the image shown below; and am trying to understand the inductive proof based on that. However, there are few issues:
Will induction be applicable, as in going from one number ($n=24$) to the next ($n+1=25$), the prime factorization differs totally from $n=24 = 2^3\cdot3$, while $n+1= 5^2$.
Why to choose strong induction is not clear, particularly when the earlier number's factorization may have nothing to deal with the next one, as shown in issue (1) above.
I hope that once the above two points are clear, the final(3rd) step of induction will also be made clear.
