CAT 2018 Progression — Previous Year Question

David has an interesting habit of spending money. He spends exactly £$X$ on the $X$th day of a month. For example, he spends exactly £$5$ on the $5$th of any month. On a few days in a year, David noticed that his cumulative spending during the last $4$ consecutive days can be expressed as $2^N$ where $N$ is a natural number. What can be the possible value(s) of $N$?