here are some clues to a mystery number.

one of my factors is 4.
when you pair me with 32, we have an LCF of 23
I am greater than 4 and less than 15

The mystery number is 8.

LCF stands for Least Common Factor, so the factors of 8 are 1, 2, 4, and 8, and the factors of 32 are 1, 2, 4, 8, 16, and 32. The only factors they have in common are 1, 2 and 4.

Therefore, the mystery number is 8.