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Math Help - integers and divisors

  1. #1
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    integers and divisors

    (a) Determine which positive integers have exactly 3 positive divisors and prove your statement.
    (b) Determine which positive integers have exactly 4 positive divisors and prove your statement.
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  2. #2
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    Hello, smithhall!

    I don't have rigorous proofs for my claims.


    (a) Determine which positive integers have exactly 3 positive divisors
    and prove your statement.

    Suppose N is the product of two distinct primes: . N \:=\:p\cdot q
    . . Then N has four divisors: . 1,\: p,\: q,\: pq

    To have exactly 3 divisors, the two prime factors must be equal: . N \:=\:p^2
    . . The divisors are: . 1,\:p,\:p^2

    Therefore, N must be the square of a prime.




    (b) Determine which positive integers have exactly 4 positive divisors
    and prove your statement.

    We answered this question in part (a).

    N must be the product of exactly two distinct primes.

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  3. #3
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    Quote Originally Posted by Soroban View Post
    We answered this question in part (a).
    N must be the product of exactly two distinct primes.
    What about N=p^3?
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