Integer division

modi4help · #1 $Old$ April 9th, 2009, 08:53 AM

Show 666 divides 1296^n - 666n + 36 for all integers n less than or equal to 1.

Opalg · #2 $Old$ April 9th, 2009, 12:01 PM

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 1296–666+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.

modi4help · #3 $Old$ April 9th, 2009, 12:21 PM

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.

Opalg · #4 $Old$ April 9th, 2009, 12:37 PM

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.