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Math Help - Is 13 divisible by this incredibly huge number?

  1. #1
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    Is 13 divisible by this incredibly huge number?

    Does 13 divide 6^13 + 33^49?

    I can't seem to think of a way to make this number more manageable.
    (2^13 * 3^13) + (3^49 * 11^49)
    3^13 can be dropped off for our purposes.

    2^13 + 3^36 * 11^49

    ...And then what. Hrm.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by BlackBlaze View Post
    Does 13 divide 6^13 + 33^49?

    I can't seem to think of a way to make this number more manageable.
    (2^13 * 3^13) + (3^49 * 11^49)
    3^13 can be dropped off for our purposes.

    2^13 + 3^36 * 11^49

    ...And then what. Hrm.
    13 is prime and so \phi(13)=12. Also, 33\equiv 7\text{ mod }13 so that 6^{13}+33^{49}\equiv 6^{13}+7^{49}\text{ mod }13. Since (13,7)=(13,6)=1 we have that 6^{13}+7^{49}=6\cdot 6^{\phi(13)}+7\cdot 7^{4\phi(13)}\equiv 6+7=13\equiv 0\text{ mod }13. So, yes it is divisible.
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