Question

Let $n\in N$ and n < $(5\sqrt{3}+8{)}^4$. Then the greatest value of n is

• A. 77040
• B. 77042
• C. 77041
• D. none of these

Answer 1

The correct option is C 77041

Suppose $x=(5\sqrt{3}+8{)}^4$
={x}+{x}
0 < $5\sqrt{3}-8$ < 1
0 < $(5\sqrt{3}-8{)}^4$
Let $F=(5\sqrt{3}-8{)}^4$
0 < F < 1
$\left[x\right]+x+F=(5\sqrt{3}+8{)}^4+(5\sqrt{3}-8{)}^4$
${=}^4{C}_0\left(5\sqrt{3}{)}^4{+}^4{C}_1\right(5\sqrt{3}{)}^3\left(8\right)+...+\left({}^4{C}_0\right(5\sqrt{3}{)}^4{-}^4{C}_1\left(5\sqrt{3}{)}^3\right(8)+...)=2.\left[C_0\right(5\sqrt{3}{)}^4+C_2\left(8{)}^2\right(5\sqrt{3}{)}^2+C_4\left(8{)}^4\right]=2\left[625\right(9)+(6\left)64\right(25\left)\right(3)+(64\left)\right(64\left)\right]=2[5625+28800+4096]=77042$
{x}+F must be an integer
Also 0 < {x}+F < 2
$\Rightarrow x+f=1$
$\Rightarrow \left[x\right]=77042-1=77041$