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Math Help - find all x

  1. #1
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    find all x

    Find all x such that
    a) 2x is congruent to x(mod5)

    (When i did this one I there is no such x, but i am not sure)

    b) -25 < x < 25 and x is congrent to 3(mod 5)

    (I get confused on the negative side, I know on the positive side you have 3, 8, 13, 18, 23 but how would it go for the negative side, would it be -3, -8, -13, -18, -23 or am i doing this wrong?)
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  2. #2
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    Quote Originally Posted by mandy123 View Post
    Find all x such that
    a) 2x is congruent to x(mod5)

    (When i did this one I there is no such x, but i am not sure)
    Subtract x from both sides.

    b) -25 < x < 25 and x is congrent to 3(mod 5)
    If x\equiv 3 ~ (5) then x = 5k+3 for some k. You want -25 < 5k +3 < 25 so -28 < 5k < 21. Thus, -5\leq k \leq 4. Put all those number into 5k+3 and you get all the x.
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  3. #3
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    Question

    ok so when I subtract x from both sides should it look like this

    2x=5k+x for some k
    2x-x=5k+x-x
    x=5k
    so then it would be x congruent to 0(mod5)?
    and then x would be every multiple of 5?
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  4. #4
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    Simply subtracting x from the congruence suffices:
    \begin{array}{rcl} 2x & \equiv & x \ (\text{mod } 5) \\ x & \equiv & 0 \ (\text{mod } 5) \qquad \text{(Subtracted x from both sides)} \end{array}

    So every x that is a multiple of 5 will satisfy your congruence. For example, take x = 20:
    2(20) = 40 \equiv 20 \ (\text{mod } 5)
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