This is an extremely interesting question. I have no idea why it works, but there is a pattern.
Here are the values of n that I found that when multiplied by 144 give 144 as the last 3 digits. Notice how the pattern repeats. 26 51 76 01. I only took it up to 1001 because I discovered that without the help of a computer program. Thanks for posting. Nice afternoon to you.
