Oh, this one is tricky. I'm going to save myself some schlepping and use Maxima to manipulate these equations. And I don't have any paper or pencils at home, so it's pretty much the only way I have of doing this.
(%i1) e1: a + b = 10;
(%o1) b + a = 10
(%i2) e2: c+d = 16;
(%o2) d + c = 16
(%i3) e3: e+f = 26;
(%o3) f + e = 26
(%i5) e4: a*p1 + b*p2 = 35;
(%o5) b p2 + a p1 = 35
(%i6) e5: c*p1 + d*p2 = 35;
(%o6) d p2 + c p1 = 35
(%i7) e6: e*p1 + f*p2 = 35;
(%o7) f p2 + e p1 = 35
(%i10) solve ( [e1,e2,e3,e4,e5,e6], [a,b,c,d,e,f,p1,p2] );
10 %r5 - 35 10 %r4 - 35 16 %r5 - 35
(%o10) [[a = -----------, b = - -----------, c = -----------,
%r5 - %r4 %r5 - %r4 %r5 - %r4
16 %r4 - 35 26 %r5 - 35 26 %r4 - 35
d = - -----------, e = -----------, f = - -----------, p1 = %r4, p2 = %r5],
%r5 - %r4 %r5 - %r4 %r5 - %r4
35 10 %r6 - 35 35 16 %r6 - 35 35 26 %r6 - 35
[a = ---, b = -----------, c = ---, d = -----------, e = ---, f = -----------,
%r6 %r6 %r6 %r6 %r6 %r6
p1 = %r6, p2 = 0]]
Ok, now we have to read the output. The bottom solution is the degenerate one, if the second price is allowed to be zero. I assume that isn't the solution we're looking for, it's the upper one, the ambiguous one. That's what those %r5 and %r4 are telling us - the thing needs more input.
Well, since a,b,c,d,e,f can only be whole numbers, and positive greater than zero, we can fix the value of two of those, and get solutions. Studying those terms real closely, it looks like c and e are the real keys here. The denominator is always negative, which requires a tight range for p2. So, we have 16*26 combinations of c and e, and the rest is brute force key search.
Here's a solution:
(%i70) e7: c = 6; e8: e = 1; solve ( [e1,e2,e3,e4,e5,e6,e7,e8], [a,b,c,d,e,f,p1,p2] );
(%o70) c = 6
(%i71)
(%o71) e = 1
(%i72)
15 5
(%o72) [[a = 9, b = 1, c = 6, d = 10, e = 1, f = 25, p1 = --, p2 = -]]
4 4
It fits all the requirements - a thru f are nonzero and whole, and p2 is less than p1. There may be other solutions, but it's getting late, and I'm sleepy.