Question

arata ca numarul n=(-4)^24-3×(-8)^15 este multiplu de 11​

Answer 1

$n = (-4)^{24} - 3\times(-8)^{15} = 4^{24} + 3\cdot 8^{15}\\ \\ n = {(2^2)}^{24} + 3\cdot {(2^3)}^{15}\\\\ = 2^{48} + 3\cdot 2^{45}\\\\ = 2^{45} (2^3 + 3)\\\\ = 2^{45}(8+3)\\\\ = 2^{45}\cdot 11 \implies n \in M_{11} \implies 11 \mid n$

Answer 2

$n = (-4)^{24}-3\cdot (-8)^{15} \\ \\ n = 4^{24}+3\cdot 8^{15} \\ \\ n = 4^{24}+(11-8)\cdot 8^{15} \\ \\n = 4^{24}+11\cdot 8^{15}-8^{16}\\ \\n = 4^{24}+M_{11}-8^{16}\\ \\ n = 2^{48}+M_{11}-2^{48} \\ \\ n = M_{11}$

⇒ n este multiplu de 11.