Let N = 2^5 3^4 5^3 . If a divisor of N is chosen at random, what is the probability that the chosen divisor is divisible by 12 but not divisible by 15 ?

Step-by-step solution

The prime factorization of N is 2^5 3^4 5^3 . The total number of divisors of N is given by the product of one more than each of the exponents of its prime factors: n(S) = (5 + 1)(4 + 1)(3 + 1) = 6 5 4 = 120 . Let E be the event that the chosen divisor is divisible by 12 but not by 15 . First, find the number of divisors divisible by 12 . Since 12 = 2^2 3^1 , a divisor 2^a 3^b 5^c is a multiple of 12 if a 2 and b 1 . The number of choices for a is 4 (which are 2, 3, 4, 5 ). The number of choices for b is 4 (which a