Limit the number of the results returned by Solve to some specific number

Question

I have this Solve:

Solve[6x1 + 30x2 + 8x3 + 15x4 - 3x5 + 2x7 == 1325 && -5x1 + 6x4 - x5 + 7x6 + 9x7 == 800 && 10 <= x1 <= 80 && 12 <= x2 <= 146 && 15 <= x3 <= 299 && 9 <= x4 <= 30 && 16 <= x5 <= 172 && 19 <= x6 <= 672 && 8 <= x7 <= 9, {x1, x2, x3, x4,x5, x6, x7}, Integers]

It returns 164254 results, which is very long. Is there a way to limit the number of returns to 100? What I am looking for is a way to avoid situations where Solve wants to return a large amount of data and limit that to some small number, to avoid unnecessary cpu usage. FindInstance is not an options, as it keeps running until I abort:

FindInstance[6x1 + 30x2 + 8x3 + 15x4 - 3x5 + 2x7 == 1325 && -5x1 + 6x4 - x5 + 7x6 + 9x7 == 800 && 10 <= x1 <= 80 && 12 <= x2 <= 146 && 15 <= x3 <= 299 && 9 <= x4 <= 30 && 16 <= x5 <= 172 && 19 <= x6 <= 672 && 8 <= x7 <= 9, {x1, x2, x3, x4,x5, x6, x7}, Integers, 100]

Answer 1

One of the ways is as follows.

Solve[6 x1 + 30 x2 + 8 x3 + 15 x4 - 3 x5 + 2 x7 == 
1325 && -5 x1 + 6 x4 - x5 + 7 x6 + 9 x7 == 800 && 10 <= x1 <= 80 &&
12 <= x2 <= 146 && 15 <= x3 <= 299 && 9 <= x4 <= 30 && 
16 <= x5 <= 172 && 19 <= x6 <= 672 && 8 <= x7 <= 9, {x1, x2, x3, 
x4, x5, x6, x7}, Integers][[1 ;; 10]]

{{x1 -> 10, x2 -> 12, x3 -> 67, x4 -> 29, x5 -> 28, x6 -> 89, x7 -> 9}, {x1 -> 10, x2 -> 12, x3 -> 70, x4 -> 30, x5 -> 41, x6 -> 90, x7 -> 9}, {x1 -> 10, x2 -> 12, x3 -> 71, x4 -> 27, x5 -> 28, x6 -> 92, x7 -> 8}, {x1 -> 10, x2 -> 12, x3 -> 73, x4 -> 24, x5 -> 19, x6 -> 92, x7 -> 9}, {x1 -> 10, x2 -> 12, x3 -> 74, x4 -> 28, x5 -> 41, x6 -> 93, x7 -> 8}, {x1 -> 10, x2 -> 12, x3 -> 76, x4 -> 25, x5 -> 32, x6 -> 93, x7 -> 9}, {x1 -> 10, x2 -> 12, x3 -> 77, x4 -> 22, x5 -> 19, x6 -> 95, x7 -> 8}, {x1 -> 10, x2 -> 12, x3 -> 77, x4 -> 29, x5 -> 54, x6 -> 94, x7 -> 8}, {x1 -> 10, x2 -> 12, x3 -> 79, x4 -> 26, x5 -> 45, x6 -> 94, x7 -> 9}, {x1 -> 10, x2 -> 12, x3 -> 80, x4 -> 23, x5 -> 32, x6 -> 96, x7 -> 8}}

Answer 2

A really simple way to get less solutions is to make your constraints tighter. Here I change the first constraint to be 10<=x[1]<=10 instead of 10<x[1]<80, and I do the same with the constraint on x[2]

A = {{6, 30, 8, 15, -3, 0, 2},
   {-5, 0, 0, 6, -1, 7, 9}};
b = {1325, 800};
xArr = Array[x, 7];
cons = 10 <= x[1] <= 80 && 12 <= x[2] <= 146 && 15 <= x[3] <= 299 && 
   9 <= x[4] <= 30 && 16 <= x[5] <= 172 && 19 <= x[6] <= 672 && 
   8 <= x[7] <= 9;
cons[[1, -1]] = 10;
cons[[2, -1]] = 12;
sol = Solve[A . xArr == b && cons, Integers];
Length@sol
(*122*)

This is not great really, because this is a very specific region of the solutions. A random sample from the constrained solution space is probably more what you want, but I am not sure how to accomplish that.

Notice however that since this is just a linear system, we can find one solution to the problem, and any other solution will be that solution plus the matrix's null space (dotted with a vector):

(*find one solution on the integers*)
f0 = FindInstance[A . xArr == b, xArr, Integers] // Values // Flatten;
(*any other solution will be f0 plus a vector dotted with the null \
space*)
ns = NullSpace[A];
kArr = Array[k, Length@ns];
solnForm = kArr . ns + Threaded[f0];
(*since the null space has 5 vectors we reduce to a 5 variable \
constrained solution*)
constrained = 
 Table[cons[[i]] /. xArr[[i]] -> solnForm[[i]], {i, Length@solnForm}]
({10 <= 135 k[1] + 35 k[2] - 10 k[3] + 60 k[4] <= 80, 
 12 <= -32 k[1] - 7 k[2] + 7 k[3] - 37 k[4] - 4 k[5] <= 146, 
 15 <= 15 k[5] <= 299, 9 <= 50 k[4] <= 30, 16 <= 7 + 50 k[3] <= 172, 
 19 <= -750 + 25 k[2] <= 672, 8 <= 673 + 75 k[1] <= 9})

Maybe there is some way to use this to find more randomly-spaced solutions in your constrained solution space?