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Math Help - Solving for m and n

  1. #1
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    Solving for m and n

    DO there exist positive integers m and n so that 76^m-34^n=10398457338?

    I am looking for m and n that would solve this equation. I know that I need to work mod 5. So 76 is 1 (mod 5) and 34 is 4 (mod 5) and 10398457338 is 3 (mod 5) but how do i really solve for m and n,
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  2. #2
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    Quote Originally Posted by chadlyter View Post
    DO there exist positive integers m and n so that 76^m-34^n=10398457338?

    I am looking for m and n that would solve this equation. I know that I need to work mod 5. So 76 is 1 (mod 5) and 34 is 4 (mod 5) and 10398457338 is 3 (mod 5) but how do i really solve for m and n,
    Note,
    76=2*38
    34=2*17
    And m<n otherwise we have a negative number.

    2^m*38^m-2^n*17^n=1039845338

    Thus, the left hand side is divisible by 2^m
    But the right hand side is divisible at most by 2 (not by 4=2^2).

    Thus, this forces m=1.
    In that case you can attempt to solve this equation for n since you determined what m has to be.
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  3. #3
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    Right!!

    I believe you are correct in saying this. I do not believe 34 to any power will give you that large number. I knew that 76 to the m power had to be 76 because of working in mod 5 will give 76 to the first power (mod 5) every time. I don't think that m and n can be found to solve this equation
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  4. #4
    Senior Member ecMathGeek's Avatar
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    Sorry to bring this up late after the discussion has ended, but I was reading through the forums and came across this problem.

    I was a little confused by the proposed solution:

    According to ThePerfectHacker, "m<n otherwise we have a negative number." I don't see how that's true.

    If m<n and, according to the example ThePerfectHacker gave, m=1, then we get:
    76^1-34^n=10398457338
    76-34^n=10398457338
    -34^n=10398457262 ... but that gives us a negative number.

    I think it's more likely that n<m, in which case, using the same analysis:
    2^m*38^m-2^n*17^n=1039845338

    This is divisible by 2^n, but 1039845338 is only divisible by 2, therefore n=1.
    76^m-34^1=10398457338
    76^m-34=10398457338
    76^m=10398457372

    However, there is no value of m that gives this solution since 'the one's place' of any solution to 76^m where m is an element of the whole numbers will always be 6.
    76^1=76
    76^2=5776
    76^3=438976
    76^4=33362176

    No solution can end in the number 2, such as 10398457372.

    I should note that I'm not a student in Number Theory and have never taken the class so if I'm wrong in all the assumptions I've made, I apologize. I'm just curious to know if my assumptions are correct.
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