Consider the problem equality, mod 5

5^b mod 5 = 0

3^a and 7^c mod 5 depend on a and c, mod 4, as follows:

a mod 4 3^a mod 5

--------- ------------

0 3

1 4

2 2

3 1

c mod 4 7^c mod 5

--------- -------------

0 2

1 4

2 3

3 1

so, the only values (a mod 4, c mod 4) which can satisfy the equality are: (0,0), (1,2), (2,3)

There might be other considerations which limit this further, but I haven't figured them out yet.