An external theorem on divisors of a number

Main result: Let ? be a collection of divisors of N = pe11?penn (e1 = minei for all i? {1,…, n}) such that if any two numbers in ? are coprime then their common multiple is N and maximal with respect to this condition, i.e. there is no other collection satisfying this condition which properly contains ?. We prove that ? consists of at least $\text{e}{}_1\text{П}{}^{\mathrm{n}}{}_{\mathrm{i\; =\; 2}}\text{(e}{}_{\mathrm{i}}\text{+ 1)?(e}{}_1\text{? 2)}$ elements.