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Math Help - Least possible value of a+b

  1. #1
    MHF Contributor alexmahone's Avatar
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    Least possible value of a+b

    Find the least pssible value of a + b, where a, b are positive integers such that 11 divides a + 13b and 13 divides a + 11b.
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    Quote Originally Posted by alexmahone View Post
    Find the least possible value of a + b, where a, b are positive integers such that 11 divides a + 13b and 13 divides a + 11b.
    I make the answer 28 (a=23, b=5), but I don't have a neat argument to prove that it's minimal.
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    Quote Originally Posted by Opalg View Post
    I make the answer 28 (a=23, b=5), but I don't have a neat argument to prove that it's minimal.
    here's a proof:

    a+13b \equiv 0 \mod 11 and a+11b \equiv 0 \mod 13 are equivalent to say that a+2b=11k and a-2b=13 \ell, for some integers k, \ell. clearly we must have: k > 0. we also get:

    2a=11k + 13 \ell, \ \ 4b=11k - 13 \ell \equiv -k - \ell \mod 4 . so: k + \ell = 4m. hence: \ell=4m - k, which gives us: a=26m - k, \ b=6k - 13m. clearly a>0, \ k > 0, implies m > 0.

    if m \geq 2, then since b > 0, we will have: k > \frac{13m}{6} > 4. thus: a+b=13m + 5k > 13 \times 2 + 5 \times 4=46 > 28.

    if m=1, then a+b=5k+13. also from b > 0, we get: k > \frac{13}{6} > 2. thus the minimum possible value of k is 3. therefore: a=26 - k = 23, \ b = 6k - 13 = 5, \ a+b=28. \ \ \ \Box
    Last edited by NonCommAlg; October 18th 2008 at 10:51 PM.
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