4!
Sorry, she didn't ask for the answer, just the question, then I had to take a nap!
The
minimum number of cuts needed to be made is 4 for a chain with 119 links. If the links are numbered serially from 1 to 119, then the cuts would be made on the following links: 6, 17, 38, and 79.
Start paying with the cut links, 1+1+1+1, at the end of the fifth week, the 5-link piece is given and the 4 one-link pieces are taken back; at the end of the tenth week, the 10-link piece is given and the 4 one-link pieces as well as the 5-link piece are taken back, and so on.