For a positive integer n, let Ω(n) denote the number of prime factors of n counted with multiplicity (e.g., Ω(3) = 1, Ω(6) = 2, Ω(9) = 2). Let p > 3 be a prime number and let N = p(p + 2)(p + 4). Which of the following statements is true?

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Mathematics · Number Theory

For a positive integer n, let Ω(n) denote the number of prime factors of n counted with multiplicity (e.g., Ω(3) = 1, Ω(6) = 2, Ω(9) = 2). Let p > 3 be a prime number and let N = p(p + 2)(p + 4). Which of the following statements is true?

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p can never be the smallest prime divisor of N

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Answer 1

Ω(N) ≥ 4

Answer 2

There exist primes p > 3 such that Ω(N) = 3

Answer 3

p can never be the smallest prime divisor of N

Answer 4

p can be the smallest prime divisor of N

For a positive integer n, let Ω(n) denote the number of prime factors of n counted with multiplicity (e.g., Ω(3) = 1, Ω(6) = 2, Ω(9) = 2). Let p > 3 be a prime number and let N = p(p + 2)(p + 4). Which of the following statements is true?

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For a positive integer n, let Ω(n) denote the number of prime factors of n counted with multiplicity (e.g., Ω(3) = 1, Ω(6) = 2, Ω(9) = 2). Let p > 3 be a prime number and let N = p(p + 2)(p + 4). Which of the following statements is true?

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