On the simultaneous equations $\sigma (2^a)=p^{f_1}{q}^{g_1},\sigma (3^b)=p^{f_2}{q}^{g_2},\sigma (5^c)=p^{f_3}{q}^{g_3}$

We shall solve the simultaneous equations $\sigma(2^a)=p^{f_1}q^{g_1}, \sigma(3^b)=p^{f_2}q^{g_2}, \sigma(5^c)=p^{f_3}q^{g_3}$with$p, q$ distinct primes.