For positive integers , call \emph{cyclically divisible} if
are all integers. Let be the set of such triples with . Determine .
[THOUGHT] We must count triples (a,b,c) with 1<=a,b,c<=5 such that (a+1)/b, (b+1)/c, (c+1)/a are all integers. I'll brute-force all 5^3=125 triples and check divisibility conditions using Python. [/THOUGHT] [PYTHON] count = 0 triples = [] for a in range(1,6): for b in range(1,6): for c in range(1,6): if (a+1) % b == 0 and (b+1) % c == 0 and (c+1) % a == 0: count += 1 triples.append((a,b,c)) print("count:", count) print("triples:", triples) [/PYTHON]
[THOUGHT] Using the brute-force Python enumeration over all triples (a,b,c) with 1≤a,b,c≤5 that satisfy the divisibility conditions, we found exactly 10 such cyclically divisible triples. Thus, |𝒯|=10. \boxed{10}