Solve system of equations with two unknowns (should be the smallest rational number possible)

I have to variables a and b

a*b = 25 = k

a and b should be the smallest integers possible, (therefore the result I should get from matlab is a = 5 and b =5, a=b will not always be the case, will depend of the value of k)

When I solve it this way I get the following:

k = 25

syms a b

eqn1 = a*b == k;

N_Results2 = solve(eqn1,[a b],'Real',true);

N_Results2 =

struct with fields:

a: 25

b: 1

How do I constraint it so that both numbers are integers and they also are the smallest numbers possible.

Answer 1

[Torsten]

Easy for a,b real:

a = b = sqrt(k).

More involved for a,b rational.

For a,b real it can be solved as an optimization if you don't believe that a=b=sqrt(k) solves the problem:

min: a + b

s.c. a*b = k

k = 5;
x0 = [1 k];
fun = @(x)x(1)+x(2);
sol = fmincon(fun,x0,[],[],[],[],[],[],@(x)nonlcon(x,k))

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

sol = 1×2

2.2361 2.2361

function [c,ceq] = nonlcon(x,k)
  ceq = x(1)*x(2) - k;
  c = [];
end

[Valeria]

I made mistake when phrasing the question, can you please take a look at it again. Thank you

[Torsten]

I assume you want to find integers a and b with a*b = k and a+b as small as possible.

You can change this to max(a,b) as small as possible or whatever you like in the code.

k = uint64(4598435343312);
v = factor(k).'

v = 7×1

2 2 2 2 3 3 31933578773

n = numel(v);
if n == 1
  disp('n is prime')
  return
end
s = 2^n;
result = logical(dec2bin(0:s-1)-'0');
f1 = 1;
f2 = k;
minimum = k+1;
for i = 2:size(result,1)-1
  factor1 = prod(v(result(i,:)));
  factor2 = k/factor1;
  if factor1 + factor2 < minimum
    minimum = factor1 + factor2;  
    f1 = factor1;
    f2 = factor2;
  end
end
minimum

minimum = uint64 31933578917

factor1

factor1 = 144

factor2

factor2 = uint64 31933578773