Let N = 2^3 3^2 5^2 7 11 13 . The number of ordered 4-tuples (p₁, p₂, p₃, p₄) of prime numbers such that the product p₁ p₂ p₃ p₄ divides N is _ _ _ _ _ _ .

Step-by-step solution

The prime factorization of N gives the multiset of available prime factors: 2 (3 times), 3 (2 times), 5 (2 times), 7 (1 time), 11 (1 time), 13 (1 time). There are 6 distinct prime factors in total. Forming an ordered 4-tuple (p₁, p₂, p₃, p₄) whose product divides N is equivalent to forming 4-letter words from this multiset of primes. We divide this into mutually exclusive cases based on the repetition of primes in the 4-tuple: Case 1: All 4 primes are distinct. We choose 4 primes from the 6 distinct available prime