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Math Help - Existence of Integers

  1. #1
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    Existence of Integers

    Suppose that a,b are two integers with gcd(m, n) = 1. Prove that there exist integers m, n
    such that a^{m} + b^{n} \equiv 1 \: mod \: ab.
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    You can split this into two easier problems: find m such that a^m\equiv1\!\!\!\pmod b, and find n such that b^n\equiv1\!\!\!\pmod a.
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    Not getting

    Hi--

    I am not getting it. I can only think of the group Z/bZ ....
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    Quote Originally Posted by Chandru1 View Post
    Suppose that a,b are two integers with gcd(m, n) = 1. When I saw the question before, I read this as gcd(a,b)=1. Presumably that what is meant?
    Prove that there exist integers m, n
    such that a^{m} + b^{n} \equiv 1 \: mod \: ab.
    Quote Originally Posted by Chandru1 View Post
    I am not getting it. I can only think of the group Z/bZ ....
    As an illustration, here's what happens when a=5 and b=7. The equation 5^m\equiv1\!\!\!\pmod7 has the solution m=6, and the equation 7^n\equiv1\!\!\!\pmod5 has the solution n=4. Then the number 5^6+7^4 will be congruent to 1 (mod 5) and also (mod 7), and therefore also (mod 35).

    [Check: 5^6+7^4 = 18026 = 515\times35+1.]
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