Let [n] represent the greatest even integer less than n that divides n, for any positive integer n. for example, [24] = 12. what is the value of [20]

First you write down all even integers that divide 20.

Those are: 2,4,10,20

Since the request is that integer is less than n, 20 cannot be the answer.
This means that greatest even integer is 10

Answer is 10.