# Find integer values that when multiplied together equal a given value

Given a = bc, with a known integer a, is it possible to find all b and c values that are integers quickly without testing each b and c value?

As an example a = 194920496263521028482429080527, is it possible to quickly find integer values for b and c?

• In general it is not easy to find all such $b$ and $c$ (basically this is factorization problem). However once we know the prime factorization of $a$, then yes. – Anurag A May 12 at 20:15
• Ok thanks, but is there a quicker way than individually testing all integer values of b and c? – so64 May 12 at 20:19
• You need to find the prime factors – Andrei May 12 at 20:26
• Integer factorization – Robert Israel May 12 at 20:46
• For your example, here's Wolfram Alpha. I hope you were not expecting to do it by hand. – Robert Israel May 12 at 20:48

The method you mention in comments, "a quicker way than individually testing all integer values of $$b$$ and $$c$$" is similar to, but less efficient than trial division, trying each integer in $$[1,\lfloor \sqrt{n} \rfloor ]$$ as a candidate for $$b$$ and determining whether each choice of $$b$$ makes $$a/b$$ an integer or not.
For your particular example, $$194920496263521028482429080527 \\ = 289673451203483 \cdot 672897345109469$$ is the only product giving that number that does not have $$b=1$$ or $$c = 1$$.