1. Show that 21*18^(2x)+36*7^(3x) is divisible by 19 for all positive integers x

2. Find all pairs of odd integers m and n which satisfy the following equation:

m+128n=3mn

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- Jun 1st 2012, 03:17 AMsomeoneanonymousTwo algebra problems? Yes I am a little stupid and even if you can do one, that's ok!
1. Show that 21*18^(2x)+36*7^(3x) is divisible by 19 for all positive integers x

2. Find all pairs of odd integers m and n which satisfy the following equation:

m+128n=3mn - Jun 1st 2012, 05:52 AMSorobanRe: Two algebra problems? Yes I am a little stupid and even if you can do one, that's
Hello, someoneanonymous!

Quote:

$\displaystyle \text{1. Show that }\,N \:=\:21(18^{2x})+36(7^{3x})\,\text{ is divisible by 19 for all positive integers }x.$

We have: .$\displaystyle 21(18^2)^x + 36(7^3)^x$

. . . . . . $\displaystyle =\;21(324)^x + 36(343)^x $

. . . . . . $\displaystyle \equiv\;21(1)^x + 36(1)^x \pmod{19}$

. . . . . . $\displaystyle \equiv\;21 + 36 \pmod{19}$

. . . . . . $\displaystyle \equiv\;57 \pmod{19}$

. . . . . . $\displaystyle \equiv\;0 \pmod{19}$

Therefore, $\displaystyle N$ is divisible by 19.