Let x=(8√3+13)^13 and y=(7√2+9)^9.

Question

Let $x=(8\sqrt{3}+13{)}^{13}$ and $y=(7\sqrt{2}+9{)}^{9}$. If $[t]$ denotes the greatest integer $\le t$, then :

Accepted Answer

Let $x'=(8\sqrt3-13)^{13}$. Then $x-x'=2\left[{}^{13}C_1(8\sqrt3)^{12}(13)+{}^{13}C_3(8\sqrt3)^{10}(13)^3+\cdots\right]$, an even integer. Since $0<13-8\sqrt3<1$... actually $0<-x'<1$ (as $8\sqrt3\approx13.86>13$ so $x'$ is small negative), so $x-x'$ even integer means $[x]$ is even.

Similarly for $y$: $y-y'=2\left[{}^9C_1(7\sqrt2)^8(9)+{}^9C_3(7\sqrt2)^6(9)^3+\cdots\right]$ is an even integer, so $[y]$ is also even.

$[x]+[y]$ is even $+$ even $=$ even.

Solutions · 1