Question

Sabiendo que A= 12*30^n tiene doble cantidad de divisores que B= 12^n *30; hallar el valor de n.

Answer 1

A = 12 . 30ⁿ
A = 3 . 2 . 2 . (3 . 5 . 2)ⁿ      CD(A) = (n + 1 + 1)(n + 1)(2 + n + 1)
A = 3 . 2² . 3ⁿ . 5ⁿ . 2ⁿ          CD(A) = (n + 2)(n + 1)(3 + n)
A = 3ⁿ⁺¹ . 5ⁿ . 2²⁺ⁿ

B = 12ⁿ . 30
B = (3 . 2²)ⁿ . 3 . 5 . 2          CD(B) = (n + 1 + 1)(2n + 1 + 1)(1 + 1)
B = 3ⁿ . 2²ⁿ . 3 . 5 . 2           CD(B) = (n + 2)(2n + 2)(2)
B = 3ⁿ⁺¹ . 2²ⁿ⁺¹ . 5

CD(A) = 2(CD(B))
(n + 2)(n + 1)(n + 3) = (2)(n + 2)(2)(n + 1)(2)
n + 3 = 8
n = 5