To find the number of perfect cube divisors of 160,000, you need to first prime factorize 160,000 and then determine how many perfect cube divisors can be formed from those prime factors.
First, let's find the prime factorization of 160,000:
160,000 = 2^5 * 5^4
Now, to find perfect cube divisors, you need to consider the exponents of the prime factors. A perfect cube divisor will have exponents that are multiples of 3.
For the factor of 2, we have 2^5. To form a perfect cube divisor, we can use 2^3, 2^6, and so on.
For the factor of 5, we have 5^4. To form a perfect cube divisor, we can use 5^0, 5^3, and so on.
So, we have multiple choices for each prime factor, and the total number of perfect cube divisors is the product of the possibilities for each prime factor.
For 2, we have 2^3, 2^6, ... 2^(5 * 3) = 2^15
For 5, we have 5^0, 5^3, ... 5^(4 * 3) = 5^12
The total number of perfect cube divisors is:
Number of perfect cube divisors = Number of possibilities for 2 * Number of possibilities for 5
Number of perfect cube divisors = (15 + 1) * (12 + 1) = 16 * 13 = 208
So, 160,000 has 208 perfect cube divisors.