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Math Help - Integer division

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    Integer division

    Show 666 divides 1296^n - 666n + 36 for all integers n less than or equal to 1.
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  2. #2
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    Quote Originally Posted by modi4help View Post
    Show 666 divides 1296^n - 666n + 36 for all integers n less than or equal to 1.
    If you mean "... for all positive integers n less than or equal to 1," then the result is true. In fact, there is only one such integer, namely n=1, and you can easily verify that 1296666+36 = 666.

    If you allow n to be zero or negative, or if you meant "... for all integers n greater than or equal to 1," then the result is false.
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    Can you explain more how can I show it please.
    sorry i wrote the question wrong.
    Show 666 divides 1296^n - 666n + 36 for all integers n greater than or equal to 1.
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    Quote Originally Posted by modi4help View Post
    Can you explain more how can I show it please.
    sorry i wrote the question wrong.
    Show 666 divides 1296^n - 666n + 36 for all integers n greater than or equal to 1.
    Sorry, I also made a mistake, in my previous comment. The result is true for all integers n greater than or equal to 1. Here's the reason.

    Start by noticing that 1296 = 36^2 and 666=18\times37. Also, x^n-1 is always divisible by x-1, and so 1296^n - 1 is divisible by 1296-1 = 35\times37. Hence 1296^n - 666n + 36 = (1296^n -1) - 37(18n -1) is divisible by 37. But 1296^n - 666n + 36 is also divisible by 18. Therefore it is divisible by 666.
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