• (natural number) X has 4 factors, which add up to 400

    From henhanna@gmail.com@21:1/5 to All on Thu Aug 4 09:18:36 2022
    Find all positive integers X such that...

    X has 4 factors, and

    the 4 factors add up to 400



    ( 6 has 4 factors: 1, 2, 3, 6)


    ------- pls wait a few days before posting answers or hints.

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  • From Mike Terry@21:1/5 to henh...@gmail.com on Thu Aug 4 20:44:14 2022
    On 04/08/2022 17:18, henh...@gmail.com wrote:

    Find all positive integers X such that...

    X has 4 factors, and

    the 4 factors add up to 400



    ( 6 has 4 factors: 1, 2, 3, 6)


    ------- pls wait a few days before posting answers or hints.


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    Repeating problem statement:

    Find all positive integers X such that...

    X has 4 factors, and

    the 4 factors add up to 400



    ( 6 has 4 factors: 1, 2, 3, 6)

    For exactly 4 factors, X will need to have either 1 or 2 prime factors, making X have one of the
    following forms (where p,q are distinct primes):

    A) X = pq
    factors: 1,p,q,pq

    B) X = p^3
    factors: 1,p,p^2,p^3 [I nearly overlooked this possibility!]

    For (A), 400 = 1+p+q+pq = (1+p)(1+q), and testing p up to 20 there are clearly no solutions. E.g.
    p=3 passes first test: 1+p = 4 | 400, but that would make q=99 which is not prime. Nearly working
    is p=19, but that would make q=19, which is prime, but p and q need to be distinct factors of X.

    For (B) trial and error shows p must be around 7, and in fact p=7 works!! 1 + 7 + 49 + 343 = 400.

    So... only solution is X = 343.


    Mike.

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