Problem Analysis #3

[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(f"Count: {count}") print(f"Triples: {triples}") [/PYTHON]

The set of cyclically divisible triples $(a,b,c)$ with $1\le a,b,c\le 5$ is determined by checking the conditions $(a+1) \equiv 0 \pmod b$, $(b+1) \equiv 0 \pmod c$, and $(c+1) \equiv 0 \pmod a$.

Based on the Python code execution, the valid triples are:

1. $(1, 1, 1)$
2. $(1, 1, 2)$
3. $(1, 2, 1)$
4. $(1, 2, 3)$
5. $(2, 1, 1)$
6. $(2, 3, 1)$
7. $(3, 1, 2)$
8. $(3, 4, 5)$
9. $(4, 5, 3)$
10. $(5, 3, 4)$

There are exactly 10 such triples.

\boxed{10}