Finding natural numbers $a, b, c$ such that $a!=b!\times c!$

Question

Yesterday, when I was playing with numbers, I was surprised to know the following relation:

$$10!=6!\times 7!.$$

Then, I've been looking for the other solutions, but I'm facing difficulty. Then, here are my questions.

Question 1 : Does there exist the other non-trivial solution?

Question 2 : Can we get all solutions? In addition to this, can we prove that they are all solutions?

I don't know if these questions are famous. Also, I'm afraid that these questions might be very easy to solve. Anyway, I need your help.

See this MSE post and its answer http://math.stackexchange.com/questions/112670/on-the-factorial-equations-a-b-c-and-abc-d — Andreas Caranti, Sep 15 '13 at 12:35

score 1 · Accepted Answer · answered Sep 15 '13 at 12:34

1

Hint: Notice that $7$ is the largest prime $\leq 10$.

answered Sep 15 '13 at 12:34

QED

6

How exactly does this contribute to solving $a!=b!c!$. – Gerry Myerson Sep 15 '13 at 12:37
that is one of the reasons for $10\times9\times8\times7$ to be equal to $7!$. We need to search for examples so that $b$ or $c$ is the largest prime lesser than $a$, that somehow narrows down the search. – QED Sep 15 '13 at 12:43
@AbishankaSaha: Thank you very much! – mathlove Sep 15 '13 at 14:59

1 Answers1