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Math Help - Need to find a number according to this:

  1. #1
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    Exclamation Need to find a number according to this:

    Hello

    How I would have to solve it, what do you think?

    I need to find the smallest natural number X [ ] with these features:

    1) is a square of whole number

    2) is a cube of whole nuber

    3) is a fifth degree of whole number

    In the natural numbers, N, find the lowest such that half of it is a perfect square, one-third is a perfect cube, and one fifth is a perfect fifth power
    Last edited by Sott; February 23rd 2013 at 06:57 AM.
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Need to find a number according to this:

    Quote Originally Posted by Sott View Post
    Hello

    How I would have to solve it, what do you think?

    I need to find the smallest natural number X [ ] with these features:

    1) is a square of whole number

    2) is a cube of whole nuber

    3) is a fifth degree of whole number
    Hi Sott!

    Let's try a couple.
    Is either of your 3 features satisfied if we pick x=1 (I'm assuming that your \mathbb N starts with 1).
    What about x=2?
    And x=3, x=4, x=5?
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  3. #3
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    Re: Need to find a number according to this:

    You have to get a whole number.... Does 1/2 or 2/3 or 4/5 or 5/2 is a whole number? No...
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    Re: Need to find a number according to this:







    and
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  5. #5
    Super Member ILikeSerena's Avatar
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    Re: Need to find a number according to this:

    (deleted)
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  6. #6
    Super Member ILikeSerena's Avatar
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    Re: Need to find a number according to this:

    Quote Originally Posted by Sott View Post
    You have to get a whole number.... Does 1/2 or 2/3 or 4/5 or 5/2 is a whole number? No...
    Ah, it seems I misunderstood.
    These features have to be true all at the same time.


    Then 2 must divide x, 3 must divide x, and 5 must divide x.

    Therefore x is of the form 2.3.5.(product of prime numbers).

    After dividing by 2 all powers of the primes in the factorization must be even.
    So x must be of the form 2^odd . 3^even . 5^even . (other primes)^even.

    After dividing by 3 the powers must be multiples of 3.
    So x must be of the form 2^(3k) . 3^(3l+1) . 5^(3m) . (other primes)^(multiple of 3).

    After dividing by 5 the powers must be multiples of 5.
    So x must be of the form 2^(5k') . 3^(5l') . 5^(5m'+1) . (other primes)^(multiple of 5).

    To find the smallest number, we need the lowest powers.
    And the numbers 2, 3, and 5 need a power of at least 1.

    The smallest number that satisfies that is:
    x = 2^15 . 3^10 . 5^6
    Thanks from Sott
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    Re: Need to find a number according to this:

    In the natural numbers, N, find the lowest such that half of it is a perfect square, one-third is a perfect cube, and one fifth is a perfect fifth power.

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  8. #8
    Super Member ILikeSerena's Avatar
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    Re: Need to find a number according to this:

    Awww... no need to shout!
    Sorry for misunderstanding - I already rectified it with a new response.
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  9. #9
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    Re: Need to find a number according to this:

    I am not shouting
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  10. #10
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    Re: Need to find a number according to this:

    So the answer is 30,233,088,000,000? Wow! Thank you very much
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