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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 σ(2 a ) = p f 1 q g 1 , σ(3 b ) = p f 2 q g 2 , σ(5 c ) = p f 3 q g 3 with p, q distinct primes.