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Math Help - Prove two composite numbers are the same

  1. #1
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    Lightbulb Prove two composite numbers are the same

    Q: Consider two composite numbers A and B. Since A is composite we can write:

    A = (p1^v1)(p2^v2)...(pn^vn)

    Where p(i) is prime and v(i) is an integer. Suppose B is made up of the same primes, the difference is their exponent:

    B = (p1^w1)(p2^w2)...(pn^wn) : w(i) is an integer

    Show that A=B if and only if v(i) = w(i) for all i.

    -----
    A: (Necessary condition) Suppose v(i) = w(i). Clearly A=B.

    But how can one deduce v(i) = w(i) given A=B as the starting point. How does one prove the sufficient condition?

    My gratitude in advance to those bright sparks that prove this proposition.
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  2. #2
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    Re: Prove two composite numbers are the same

    Quote Originally Posted by bourbaki87 View Post
    Q: Consider two composite numbers A and B. Since A is composite we can write:

    A = (p1^v1)(p2^v2)...(pn^vn)
    Where p(i) is prime and v(i) is an integer. Suppose B is made up of the same primes, the difference is their exponent:

    B = (p1^w1)(p2^w2)...(pn^wn) : w(i) is an integer

    Show that A=B if and only if v(i) = w(i) for all i.
    Comment: it usual in composite numbers for v_i to be non-negative integers.

    Take a simple case.
    GIVEN: 2^x\cdot 3^y=2^{13}\cdot 3^{14}.
    What must x~\&~y equal and why?
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  3. #3
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    Re: Prove two composite numbers are the same

    Answer:

    The fundamental theorem of arithmetic: Every integer can be written as a unique product of primes (up to ordering of factors). Thus A=B has only one such product of primes, due to the uniqueness asserted by the F.T.A. Therefore all exponents of primes in A must be equal to the exponents of primes in B thus v(i) = w(i). Q.E.D
    Last edited by bourbaki87; November 15th 2011 at 11:05 AM. Reason: Clarification
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