18. If x, y, and z are different positive prime numbers, then the product xyz is divisible by how many different positive integers?
Answer: E
The product xyz is divisible by eight different positive integers.
The product of three different positive prime numbers x, y, and z, denoted as xyz, is divisible by eight different positive integers. This is derived from the fact that the number of divisors of a product of prime numbers can be calculated using the formula: (e1 + 1)(e2 + 1)(e3 + 1), where e1, e2, and e3 are the exponents of the prime factors in the product. Since each prime appears to the power of one in this case, the calculation is (1+1)(1+1)(1+1) = 2 * 2 * 2 = 8.
A) One
This option is incorrect because a product of three different positive primes cannot have only one positive divisor. The only number that has one positive divisor is 1 itself, which is not the case here.
B) Three
This option is also incorrect. A product of three distinct primes will have more than three divisors. Each prime contributes at least two factors (itself and one), leading to a significantly higher count of divisors.
C) Four
This option is incorrect as well. While four is a possible number of divisors for certain products, the product of three distinct primes must have a greater number of divisors, as calculated through the divisor formula.
D) Six
This option is incorrect. A total of six divisors would suggest that the product can be expressed with either fewer prime factors or some primes raised to higher powers, neither of which applies to the product xyz where all primes are distinct and raised to the first power.
E) Eight
This option is correct. As explained, the total number of divisors for the product of three distinct prime numbers is calculated as (1+1)(1+1)(1+1) = 2 * 2 * 2 = 8. This means that xyz is divisible by exactly eight different positive integers.
Conclusion
The correct answer, eight, is derived from the divisor counting formula for the product of distinct primes. All other options are incorrect as they underestimate the number of divisors for the product of three different positive prime numbers. Thus, the correct understanding of prime factorization and its implications on divisibility confirms that the product xyz is indeed divisible by eight different positive integers.