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