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Math Help - Fundamental Theorem of Arithmetic

  1. #1
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    Fundamental Theorem of Arithmetic

    Use the Fundamental Theorem of Arithmetic to show every positive integer, n>0, can be written uniquely as the product of a power of 2 and an odd number.
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  2. #2
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    Suppose that there are two distinct representations of the same number as the product of a power of 2 and an odd number (let us call them 2^{p_a}O_a and 2^{p_b}O_b. This number has a unique decomposition into primes, from which it immediately follows that p_a = p_b, since O_a and O_b contain no factors of 2, and 2 is prime. Thus 2^{p_a} = 2^{p_b} = c. Now cO_a = cO_b, and since c \geq 1, it follows that O_a = O_b, thus demonstrating that the two representations are not distinct, and hence the number has a unique representation as a power of 2 and an odd number.
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  3. #3
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    I see where your proof shows uniqueness, but I don't see how it shows that the integers can be written as the product of a power of 2 and an odd number.
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  4. #4
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    Quote Originally Posted by tarheelborn View Post
    I see where your proof shows uniqueness, but I don't see how it shows that the integers can be written as the product of a power of 2 and an odd number.
    Oh, I guess I just overlooked that part because it seemed obvious. Write a number as its unique product of primes:

    2^{p_1}3^{p_2}5^{p_3}...

    Now, using associativity, this is equal to

    2^{p_1} \cdot (3^{p_2}5^{p_3}...)

    The left term is a power of two and the right term is an odd number, because every product of only odd numbers is odd. You can prove that statement easily using modular arithmetic.
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  5. #5
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    Thank you!!!
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