Quick help for a homework question, please include a solution. You were talking with your friends at lunch comparing how the school year was going. Your friend, Kate, complained that she can’t always remember her locker combination.

You said, “What I do is try to find some pattern in the numbers. For example, my locker combination last year was 22-33-11, or eleven multiplied by 2, 3 and 1.”

“You were lucky with that locker combination. I wish I could see a pattern in my numbers,” said Kate.

“If you show me the numbers,” you answered, “I’ll try to help you think of something. We’re lucky, though, that the numbers on our locks only go up to 39!”

Kate held out a piece of paper. She said:
Two of my numbers are one-digit numbers.
The middle number is a two-digit number that is divisible by 6.
If you look at the first number and the third number, each number is one less than a factor of the middle number.
If you add 1 to the first number and also to the last number and find their product, you get the middle number.

What are all the different possibilities for Kate’s locker combination?
Note the combination requires three numbers. There are more than 10 combinations, but less than 20.