1. Difficult problem solving quesiton

Hello Mathwizards!

I have a problem solving question that is basically this-

"Find the greatest factor of 11! that is one greater than a multiple of 6"
That's all it gives you

And I think that the answer is 385. However I am unsure how to word it or show how I done it.
The way I worked it out was to use a factor calculator online to find all the factors of 11! and then subtracted 1 then divided by 6. If this gave a whole number then this is a possibility. I did this for all odd factors as I fathomed that 1 greater than a multiple of 6 must be odd as a multiple of 6 will always be even.
This seems like a tedious process and I am sure that there must be an easier method.

I tried creating equations with x=a factor of 11! and y=a multiple of 6 but the only two I could get were
x-1=y
x=y+1

I have to display full working and I dont want to say that I have done that the method above for hundreds of numbers.

Any help or even pointers in the right direction would be GREATLY appreciated.