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Math Help - not understanding

  1. #1
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    not understanding

    have to do a couple problems for homework and im not understanding exactly what is going on and why its the answer

    27. Let m, n, and d be integers. Show that if d \ m and d \ n, then d\ (m - n).

    my professor gave us the answer to this but i do not understand why it works
    answer: m = dq1 n = dq2 m - n = dq1 - dq2 = d(q1 - q2) therefore d \ (m - n)

    these are the other problems i have to do

    28. Let m, n, and d be integers. Show that if d \ m, then d \ mn.

    31. Let a, b, and c be integers. Show that if a \ b and b \ c, then a \ c.

    33. Give an example of consecutive primes p1 = 2, p2 ..., pn
    where
    p1p2... pn + 1
    is not prime.

    any help would be much appreciated
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by pooshipple View Post
    have to do a couple problems for homework and im not understanding exactly what is going on and why its the answer

    27. Let m, n, and d be integers. Show that if d \ m and d \ n, then d\ (m - n).
    If d\mid m,d\mid n then m=dz,n=dz' for some z,z'\in\mathbb{Z}. Clearly then m-n=dz-dz'=d\left(z-z'\right) and since z,z'\in\mathbb{Z}\implies z-z'\in\mathbb{Z} the conclusion follows.
    my professor gave us the answer to this but i do not understand why it works
    answer: m = dq1 n = dq2 m - n = dq1 - dq2 = d(q1 - q2) therefore d \ (m - n)
    Sorry, I just noticed this. What it means to say that d\mid n is that d divides n or that \frac{n}{d}\in\mathbb{Z}. Does that make a little more sense? So if m-n=d\left(z-z'\right) then \frac{m-n}{d}=\frac{d\left(z-z'\right)}{d}=z-z'\in\mathbb{Z}

    these are the other problems i have to do

    28. Let m, n, and d be integers. Show that if d \ m, then d \ mn.
    d\mid m\implies m=dz so then mn=dzn or \frac{mn}{d}=zn and since the integers are closed under multiplication the conclusion follows.

    31. Let a, b, and c be integers. Show that if a \ b and b \ c, then a \ c.
    These all can be done the same. a\mid b\implies b=za and b\mid c\implies c=bz'. So then c=bz'=\left(az\right)z' and the conclusion follows.


    33. Give an example of consecutive primes p1 = 2, p2 ..., pn
    where
    p1p2... pn + 1
    is not prime.
    What does "consecutive" mean? That p_2=p_1+1 or that is merely the next prime in the list of primes? I'm assuming the latter. What do you think?
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