Question 3
Let $n\gt 3$ be an integer. Choose three numbers from $\{1,2,\ldots,n\}$. Using each chosen number once and using addition, multiplication and parentheses, form all possible combinations.
(a) Show that if all three numbers exceed $n/2$, the values of these combinations are all distinct.
(b) Let $p\le\sqrt n$ be prime. Show that the number of ways to choose three numbers with smallest number $p$ for which the combination values are not all distinct equals the number of positive divisors of $p-1$.
Editorial clarification (AI): expressions differing only by commutativity or associativity are counted as the same combination.
(a) Show that if all three numbers exceed $n/2$, the values of these combinations are all distinct.
(b) Let $p\le\sqrt n$ be prime. Show that the number of ways to choose three numbers with smallest number $p$ for which the combination values are not all distinct equals the number of positive divisors of $p-1$.
Editorial clarification (AI): expressions differing only by commutativity or associativity are counted as the same combination.