For positive integers , call \emph{cyclically divisible} if
are all integers. Let be the set of such triples with . Determine .
[THOUGHT] I need to find all triples (a, b, c) with 1 ≤ a, b, c ≤ 5 such that:
- (a+1)/b is an integer (i.e., b divides a+1)
- (b+1)/c is an integer (i.e., c divides b+1)
- (c+1)/a is an integer (i.e., a divides c+1)
Let me enumerate all possible triples and check these conditions. [/THOUGHT] [PYTHON] count = 0 valid_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 valid_triples.append((a, b, c))
print("Count:", count) print("Valid triples:", valid_triples) [/PYTHON]
[THOUGHT] The Python code enumerated all triples (a, b, c) with 1 ≤ a, b, c ≤ 5 and checked the divisibility conditions. It found exactly 10 valid triples that satisfy all three conditions. [/THOUGHT]
\boxed{10}