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Solve the congruence to the smallest possible modulus.

  1. Apr 2, 2012 #1
    1. The problem statement, all variables and given/known data

    solve the congruence to the smallest possible modulus.

    38x[itex]\equiv[/itex]10 (mod84)



    2. Relevant equations



    3. The attempt at a solution

    Ok so I thought I knew how to do these but apparently not.

    So I start out by

    38x+84y = 10

    Now I think I should find the gcd so I can simplify

    so

    84 = 38(2) + 8[itex]\Rightarrow[/itex] 8= 84 - 38(2)

    38= 8(4) + 6 [itex]\Rightarrow[/itex] 6= 38 - 8(4)

    8 = 6(1) + 2 [itex]\Rightarrow[/itex] 2 = 8 - 6(1)

    6 = 2(3) + 0 [itex]\Leftrightarrow[/itex] gcd = 2

    so 19x + 42y = 5


    but then where? I seem to get lost at this point?
     
  2. jcsd
  3. Apr 2, 2012 #2

    K^2

    User Avatar
    Science Advisor

    You need to use GCD algorithm, but not just to find GCD. It's a more powerful tool. Let me demonstrate.

    Lets start with your simplified version, 19x=5 (mod42), which you turn into 19x+42y=5. But if you know x, you know y. If you know y, you know x. So we can switch things around.

    42y=5 (mod19)

    Except 42>19, and we can take 19 away twice.

    4y=5 (mod19)

    Better! But why should we stop?

    4y+19z=5

    19z=5 (mod4)

    3z=5 (mod4)

    3z+4w=5

    4w=5 (mod3)

    w=5

    Woo-hoo! Now, you can use w to find z, z to find y... I'll let you do all that.
     
  4. Apr 2, 2012 #3
    ok good got it now thanks.
     
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