Is there a function to find the paired factors of an integer?

Is there a function in Matlab to find the pairs of factors that multiply together to give a specified integer, e.g,

F=somefunc(16)

F =

1 16

2 8

4 4

Answer 1

[John D'Errico]

allfactorpairs = @(n) [divisors(n)',n./divisors(n)'];

allfactorpairs(16)

ans = 5×2

1 16 2 8 4 4 8 2 16 1

If you want only those pairs where the first element is the smaller of the two, then you might write a little function, like this:

allfactorpairs2(16)

ans = 3×2

1 16 2 8 4 4

allfactorpairs2(210)

ans = 8×2

1 210 2 105 3 70 5 42 6 35 7 30 10 21 14 15

allfactorpairs2(137)

ans = 1×2

1 137

Finally, see that this code runs on both doubles as well as syms, and is efficient as long as the number is esily factored, so the limit is really in terms of how well factor works for huge numbers. That of course can be terribly difficult.

allfactorpairs2(sym(factorial(10)))

ans = 

Your choice. Note that divisors is a function from the symbolic toolbox. So if someone finds it is too slow, or you lack that toolbox, it is easily replaced. And while I expect that @Matt J has the symbolic TB, I'll write it the replacement below to operate efficiently on doubles as well as syms.

function D = principaldivisors(N)
  % Compute the principal divisors of the scalar variable N, so only those divisors < N.
  % Note if N is prime, then this function returns ONLY 1, as the only factor less than N.
  F = factor(N);
  D = 1;
  if numel(F) > 1
    % N is composite, so it has more than one divisor less than N
    % if N were prime, then 1 and N are the only factors.
    for Fi = F
      D = unique([D,D*Fi]);
    end
    D(D == N) = [];
  end
end

function pairs = allfactorpairs2(N)
  D = principaldivisors(N)';
  D(D > sqrt(N)) = [];
  pairs = [D,N./D];
end

[Matt J]

Great! I am surprised, though, that the Symbolic Toolbox is required.

[John D'Errico]

Arguably there should be a utiltiy in MATLAB itself to do this, or even just to find a list of all proper divisors of an integer. But there are relatively few utilities in MATLAB for working with integers, not much more than primes, factor, gcd, lcm, and a few others.

Should there be an expansion of those tools? Perhaps.