Find the number of different ways in which 711 can be expressed as the product…

Concept: To express a prime power pn as a product of three positive factors, where the order of the factors does not matter, split the exponent n among the three factors as px . py . pz with x + y + z = n and x, y, z >= 0 (a factor of 1 corresponds to an exponent of 0). The number of such unordered triples (x, y, z) equals the number of partitions of n into at most three parts.

Application: Here the number is 711, so n = 11. Count every partition of 11 into at most three parts, grouped by how many parts are used:

1. One part: {11} -- 1 way (the three factors are 711, 1, 1).

2. Two parts: (10, 1), (9, 2), (8, 3), (7, 4), (6, 5) -- 5 ways.

3. Three parts: (9, 1, 1), (8, 2, 1), (7, 3, 1), (7, 2, 2), (6, 4, 1), (6, 3, 2), (5, 5, 1), (5, 4, 2), (5, 3, 3), (4, 4, 3) -- 10 ways.

4. Adding the three cases: 1 + 5 + 10 = 16.

Cross-check: The number of partitions of n into at most three parts is also given by the nearest-integer formula round((n + 3)2 / 12). For n = 11 this gives round(142 / 12) = round(196 / 12) = round(16.33) = 16, matching the direct count above.

So 711 can be expressed as a product of three factors in 16 distinct ways.