Question

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

Accepted Answer

B. Sum of 4 consecutive days in same month = $d + (d+1) + (d+2) + (d+3) = 4d + 6$. Setting $4d + 6 = 2^N$: - $N=5: 4d+6=32 \implies 4d=26$ (not integer). - $N=6: 4d+6=64 \implies 4d=58$ (not integer, but across two month boundaries: e.g., 30th, 31st, 1st, 2nd gives $30+31+1+2 = 64 = 2^6$). This occurs in 6 months of a year. Thus $N = 6$ is uniquely possible.

Consecutive Day Sums as Powers of 2

Explanation

Practise this under exam conditions