I'm trying to solve an equation of this type on integers
(10 - x) (6 - y) (4 - z) (3 - w) == x*y*z*w
for
1 <= x <= 10
1 <= y <= 6
1 <= z <= 4
1 <= w <= 3
I tried Solve and Reduce and they work fine
Reduce[(10 - x) (6 - y) (4 - z) (3 - w) == x*y*z*w
&& 1 <= x <= 10 && 1 <= y <= 6 && 1 <= z <= 4 && 1 <= w <= 3, {x, y, z, w}, Integers]
Solve[(10 - x) (6 - y) (4 - z) (3 - w) == x*y*z*w && 1 <= x <= 10 &&
1 <= y <= 6 && 1 <= z <= 4 && 1 <= w <= 3, {x, y, z, w}, Integers]
they both give the right answer
The problem is that I need to Solve bigger equations (of the same type) with more variables like
ClearAll[x, y, z, w, a, b, c, d]
Solve[(20 - x) (10 - y) (6 - z) (4 - w) (3 - a) (3 - b) (3 - c) (3 - d) == x*y*z*w*a*b*c*d
&& 1 <= x <= 20 && 1 <= y <= 10 &&
1 <= z < 6 && 1 <= w < 4 && 1 <= a <= 3 && 1 <= b <= 3 &&
1 <= c < 3 && 1 <= d < 3, {x, y, z, w, a, b, c, d}, Integers]
but even the above one (with 8 variables) takes forever (this one should return 272 solutions)
In fact I want to solve an equation with more than 20 variables
What is the most efficient way to solve equations like these?
I want the "number of solutions" of this one
Reduce[(99 - a) (50 - b) (26 - c) (18 - d) (11 - e) (9 - f) (7 -
g) (7 - h) (6 - i) (5 - j) (5 - k) (4 - l) (4 - m) (4 - n) (4 -
o) (3 - p) (3 - q) (3 - r) (3 - s) (3 - t) (3 - u) (3 - v) (3 -
w) (3 - x) (3 - y) ==
a*b*c*d*e*f*g*h*i*j*k*l*m*n*o*p*q*r*s*t*u*v*w*x*y && 1 <= a <= 99 &&
1 <= b <= 50 && 1 <= c <= 26 && 1 <= d <= 18 && 1 <= e <= 11 &&
1 <= f <= 9 && 1 <= g < 7 && 1 <= h < 7 && 1 <= i <= 6 &&
1 <= j <= 5 && 1 <= k <= 5 && 1 <= l <= 4 && 1 <= m <= 4 &&
1 <= n <= 4 && 1 <= o <= 4 && 1 <= p <= 3 && 1 <= q <= 3 &&
1 <= r <= 3 && 1 <= s < 3 && 1 <= t <= 3 && 1 <= u <= 3 &&
1 <= v <= 3 && 1 <= w < 3 && 1 <= x < 3 && 1 <= y < 3, {a, b, c, d,
e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x,
y}, Integers]