Finding the Product of Primes: A Number Theory Puzzle

[evansmiley]

Hi i found a question in number theory, involving two equations, it goes as follows:
Let p1, p2, p3 and p4 be 4 different prime numbers satisfying the equations
2p1 + 3p2 + 5p3 + 7p4 = 162
11p1 + 7p2 + 5p3 + 4p4 = 162
Find all possible values of p1p2p3p4.

Not knowing what to do, i used the fact that even plus odd numbers add to give odd numbers to deduce that one of either p2 or p3 is 2. Also, by taking the two equations from each other, and some inequalities i managed to break down p1 and p4 into sets of possible prime numbers (p1 was possibly 3,5,7 or 11, and p4 was either 13, 17 or 10) and i was forced into testing each value of p4 and seeing if other solutions are possible, however there was only one possible answer which was p1 = 5, p2 = 3, p3 = 2, and p4 = 19, which gives the product value of 570.
Two questions - is this actually right ? and secondly, surely there's a much nicer way to find the answer to this question, maybe one which is more indirect seeing as we must only find the possible values of the product, not the values of the prime numbers themselves?
EDIT: Sorry just saw that rule about no posting of any "homework-style" questions, sorry.

 

[Norwegian]

Yes, you are right. My way was to first subtract the equations, immediately giving p2=3. Then p3=2 as you say, and we are left with 2 linear eqs in 2 unknowns, giving us 5 and 19. I don't think there is a natural way of only getting the product, without getting the primes.