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A positive integer n is written as a product of prime factors. How many distinct prime factors does n have?
(1) n has exactly 8 positive divisors.
(2) n is divisible by 30.
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Correct answer
C
The number of divisors is the product of (each prime's exponent plus 1) over the distinct primes.
Statement (1): 8 divisors factors as 8, or 4 × 2, or 2 × 2 × 2, corresponding to one prime raised to the 7th power (1 distinct prime), one prime cubed times another prime (2 distinct primes), or three distinct primes each to the first power (3 distinct primes). The distinct-prime count is 1, 2, or 3. not sufficient.
Statement (2): divisible by 30 = 2 × 3 × 5 forces at least 3 distinct primes, but n could have 3, 4, or more. not sufficient.
Together: n has exactly 8 divisors and at least 3 distinct primes. The only way to write 8 as a product of 3 or more factors each at least 2 is 2 × 2 × 2, which corresponds to exactly three distinct primes. So n has exactly 3 distinct primes. Determinate. Sufficient; neither alone.
The (E) lure misses that requiring at least 3 primes pins the divisor structure to 2 × 2 × 2, fixing the count at exactly 3.
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