Is it possible to solve analytically this problem? "To find 4 integers 10<X,Y,Z,W<20 so that X*Y+Z*W=539"

Question

I would like to know the methodology applied to solve this kind of problems

Answer 1

This problem can be solved by a sophisticated search.
Let's look at z*w. Since z < = 19 and w <= 19 let's
set z = 11. Now try all possible values for w.
For each such pair compute 539 - zw and
factor the resulting number. Check if it has
two factors in the given range.
Now increase z by 1 and repeat . Do this
step from z = 12 to 19.
Note that it may seem as if there are 81 pairs
to try, but in fact there are only 45, because
if the pair zw does not yield a solution, neither
does wz.
I tried all pairs with z = 11 and none yielded
a solution, but z= 12 worked:
The pair 12, 18 yields xy = 323 = 17*19.
So a solution is 17*19 + 12*18.
I don't know if this is the only solution, as I
didn't check further.

Answer 2

Excel solver can solve this problem once the appropriate constraints are entered. For X, Y, Z, and W there are three constraints each.

1. All four are integers
2. All four are greater than or equal to 11, and
3. All four are less than or equal to 19.

You want the answer to be exactly 539. Just plug in the numbers and stir gently.

voila 12*18 + 17*19 = 539

Answer 3

Analytically, I don't think it can be solved without additional information. Your best bet for something like this is Mathcad or other math software package. If that is not an option, you could brute force it with some simple code (PERL for example --- which is free). By the way... there is no solution.

$counter = 0;
$X = 11;
$Y = 11;
$Z = 11;
$W = 11;
$FN = 0;
while ($FN != 539)
{
while ($X < 20)
{
while ($Y < 20)
{
while ($Z < 20)
{
while ($W < 20)
{
$FN = $X * $Y + $Z * $W;
$counter ++;
$W ++;
};
$W = 11;
$Z ++;
};
$Z = 11;
$Y ++;
};
$Y = 11;
$X ++;
};
$X = 11;
};
};

Answer 4

Perhaps it is impossible to find analytical integer methods for such quadratic problems. But you can reduce it somewhat to form simpler sub-problems, and solve it by say, brute-force.

Let A = X*Y, and let B=Z*W

Then A = 539 - B

You don;t seem to have a lower bound. Assuming you have one, say at 0, then form a box for A as follows: ((0,0), (0,20),(10,0),(10,20)), similarly for B. On a line, A takes values in [0,200] = 201 integer values. Similarly do so for B. For each value from A's line get a value in th B line. Form a table.

Then for each value in A (=X*Y), get X and Y for integer values from inside A's box. Similarly for B.

Answer 5

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Answer 6

Yes . Use Operation research.
In that u need to use simplex method.
If u notice it, there are 5 equations here, considering the < and > .

Answer 7

no, i dont think u can solve this problem
since u have 4 variables (w,x,y,z), u need 4 equations to solve these 4 variables

Answer 8

only by hit & trick method

Answer 9

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