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    Well-Ordering and Division Theorem

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    32. Show that there is no rational number b/a whose square is 2, as follows: if b^2 = 2a^2, then b is even, so b = 2c, so, substituting and cancelling 2, 2c^2 = a^2. Use that argument and well-ordering to show that there can be no natural number a > 0 with b^2 = 2a^2 for some natural number b.

    33. Let m be the least common multiple of a and b, and let c be a common multiple of a and b. Show that m divides c. Hint: use the division theorem on m and c, and show that the remainder r is a common multiple of a and b, hence r = 0.

    © BrainMass Inc. brainmass.com September 27, 2022, 4:29 pm ad1c9bdddf
    https://brainmass.com/math/number-theory/well-ordering-division-theorem-524474

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    Let's us review the concept of well ordering principle: if a subset of integers is bounded below, then it has a minimal element. With this in mind, let us do these problems:

    32. Suppose otherwise, let b/a = √2, then b2 = 2a^2. The given argument tell us
    you can always find b = 2c such that a^2 = 2c^2. The trick is to notice that a/c = √2, and a < b.

    Let us use well ordering principle: choose b/a = √2 such that b is the smallest (please think about why we can apply the well ordering principle: think about the following set {x ∈ N : x/y =√2}, is it a subset of integers? is it bounded below?). Then apply the given argument, we find another c such that a/c = √2 where a < b. This contradicts our assumption that b is the smallest.

    33. Division algorithm says: given x, p, where p 6= 0, there exists a, b such that x = a · p + b, and 0 ≤ b < p. Here, x, p, a, b are all integers. Now given m, c apply division algorithm to get c = x · m + r where 0 ≤ r < m.
    Suppose m does not divide c, then r 6= 0. Rewrite it as r = c − x · m, where x is an integer. Now a, b both divides c, m. Think about why it follows that a, b also divides r.

    Now m is the least common multiple of a, b, r is another common multiple of a, b with 0 < r < m. So we get a smaller common multiple which gives us a contradiction (Recall the least common multiple is the smallest positive natural number that can be divided by both a, b).

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com September 27, 2022, 4:29 pm ad1c9bdddf>
    https://brainmass.com/math/number-theory/well-ordering-division-theorem-524474

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