CMSC 250 Quiz #5 KEY Monday, Oct. 8, 2001

Write all answers legibly in the space provided. The number of points possible for each question is indicated in square brackets - the total number of points on the quiz is 30, and you will have exactly 15 minutes to complete this quiz. You may not use calculators, textbooks or any other aids during this quiz.
  1. [20 pnts.] Disprove by counter example or Prove each of the following:

    1. The sum of any rational number and any integer is rational.





      show: $\forall r \in Q, \forall m \in Z, r+m \in Q$
      PROOF:
      USE:
      $r \in Q \rightarrow \exists a,b \in Z where b \neq 0, r = \frac{a}{b}$
      ALGEBRA:
      $r + m = \frac{a}{b} + m$
      $\,\,\,\,\,= \frac{a}{b} + \frac{mb}{b}$
      $\,\,\,\,\,= \frac{a+mb}{b}$

      $a+mb \in Z$ because of closure of + and * over Z
      $b \in Z$ by definition
      therefore $r+m \in Q $ becuase it is a quotient of integers



    2. If $n$ is an odd integer, then $n^2 \equiv 1 \,\,\,mod \,\,\,2$.





      PROOF:
      USE:
      $n \in Z^{odd} \rightarrow \exists k \in Z, n = 2k + 1 $
      ALGEBRA:
      $n = 2k+1$
      $n^2 = (2k+1)^2$
      $\,\,\,\,\,\,= 4k^2 + 4k + 1$
      $\,\,\,\,\,\,= 2(2k^2 + 2k) + 1$

      $ n^2 \in Z^{odd}$ because $(2k^2 + 2k) \in Z$

      therefore $n^2 \equiv 1 $ mod 2.



    Grading:
    10 points for each question
    divided up into:
    5 points for knowing the definition to apply (i.e. rational = a/b)
    4 points for algebra of getting from there to the conclusion
    1 point for form and statement of the conclusion

  2. [4 pnts.] Write the standard factored form of 1050:

    $1050 = 2^1*3^1*5^2*7^1$

    Grading:
    2 points for knowing what stanard factored form is
    2 points for knowing the prime factors of 1050

  3. [6 pnts.] Use the unique factorization theorem and suppose that m is an integer such that

    \begin{displaymath} 5 * 4 * 3 * 2 * m = 10 * 11 * 12 * 13 \end{displaymath}

    Circle Yes or No for each of the following: Yes means that this is something that must be true, No means it doesn't necessarily need to be true:

           
    a) $10 \vert m$ YES (NO)
           
           
    b) $11 \vert m$ (YES) NO
           
           
    c) $12 \vert m$ YES (NO)
           
           
    d) $13 \vert m$ (YES) NO
           
           
    e) $24 \vert m$ YES (NO)
           
           
    f) $143 \vert m$ (YES) NO
           

    Grading:
    1 point per line

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John Arras
2001-10-10