Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 2 / a

Use strong induction with three consecutive base cases:For $n\ge 11$, assume all values from $8$ through $n-1$ have the required representation. Since $8\le n-3<n$, write $n-3=3k+5m$ with $k,m$ nonnegative integers. Then $n=3(k+1)+5m$ is another such representation. Strong induction proves the claim for every $n\ge 8$. This is a consecutive-base induction for additive semigroups: the three base cases handle all residues before repeated addition of $3$.