Question
Show that 81^7-27^9-9^{13} is divisible by 45
Show that 81^7-27^9-9^{13} is divisible by 45
81^7-27^9-9^{13}
=(9^2)^7 - (3\times9)^9-9^{13}
=9^{14} - 3\cdot 3^8\cdot 9^9-9^{13}
=9^{14} - 3\cdot 9^4\cdot 9^9-9^{13}
=9^{14} - 3\cdot 9^{13}-9^{13}
=9^{14} - 4\cdot 9^{13}
=9^{13}(9-4)
=5\cdot9^{13}
Therefore
81^7-27^9-9^{13} is divisible by 45