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Math Help - 2 More Questions

  1. #1
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    Mar 2009
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    2 More Questions

    1. Find all solutions, if any, to the system of congruences.

    x = 5 (mod 6)
    x = 3 (mod 10)
    x = 8 (mod 15)

    I tried following an example out of my book and got the following, which I'm fairly sure is horribly wrong:

    m = 900; M1 = 150; M2 = 90; M3 = 60
    150 = a mod 6; a = 12;
    90 = b mod 10; b = 20;
    60 = c mod 15; c = 30;

    z = 5*150*12 + 3*90*20 + 8*60*30
    z = 28,800

    z = x (mod m)
    28,800 = x (mod 900)
    so... x = 900. This, doesn't seem right, any help would be appreciated.

    2. Use Strong induction to show that every positive integer n can be written as a combination of distinct powers of 2. Ex. 2^0 = 1, 2^1 = 2, 2^2 = 4.

    Here's what I have. I think I have proven it, but I'm not sure if it's by strong induction.

    Basis: 2^0 = 1

    Inductive:
    Assume: 2^x + 2^y = j
    We must show 2^x + 2^y = (j+1)

    if (j+1) is odd. x = (j+1)/2 rounded down. y = 0.
    /* ie. for j = 2, x = 1, y = 0; */

    if (j+1) is even. x = (j+1)/2 - 2, y = 1;

    Thus we have shown that the basis and inductive steps are true.
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  2. #2
    Super Member
    Joined
    Jan 2009
    Posts
    591
    Quote Originally Posted by Niotsueq View Post
    1. Find all solutions, if any, to the system of congruences.

    x = 5 (mod 6)
    x = 3 (mod 10)
    x = 8 (mod 15)
    As a very quick check, you should recognize that the one's digit must be a 3. Your number x must end with a three.

    x = 3 mod 10
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