Is pq = 1? (1) pqp = p (2) qpq = q

Notice that p=0 from the first statement does not exclude q being 0 too and similarly q=0 from the second statement does not exclude p being 0 too.

Is pq = 1?

(1) pqp = p --> p(pq-1)=0 --> either p=0 and q=(any number), including 0 so in this case pq=0 not 1 or pq=1. Not sufficient.

(2) qpq = q --> q(pq-1)=0 --> either q=0 and p=(any number), including 0 so in this case pq=0 not 1 or pq=1. Not sufficient.

(1)+(2) When combined we have that either p=q=0 or pq=1. Not sufficient.

Answer: E.

When we consider two statements together we should take the values which satisfy both statements. For this question \(pq=1\) satisfies both statement, but \(p=q=0\) also satisfies both statements. So what you call "common solution" for this question is: \(pq=1\) OR \(pq=0\neq{1}\).

Hope it's clear.