If [x] denotes the greatest integer function | vedclass

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(D) Let $x = (2+\sqrt{3})^5$ and $y = (2-\sqrt{3})^5$. Since $0 < 2-\sqrt{3} < 1$,we have $0 < y < 1$. Consider the expression $S = (2+\sqrt{3})^5 + (

Vedclass · Accepted Answer

$8$

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(D) Let $x = (2+\sqrt{3})^5$ and $y = (2-\sqrt{3})^5$. Since $0 Consider the expression $S = (2+\sqrt{3})^5 + (2-\sqrt{3})^5$. Using the binomial expansion,the irrational terms cancel out,leaving $S$ as an even integer. $S = 2 	imes [^5C_0 \cdot 2^5 + ^5C_2 \cdot 2^3 \cdot 3 + ^5C_4 \cdot 2^1 \cdot 3^2] = 2 	imes [32 + 80 	imes 3 + 10 	imes 18] = 2 	imes [32 + 240 + 180] = 2 	imes 452 = 904$. Since $S = x + y = 904$ and $0 Thus,$[x] = 904 - 1 = 903$. Now,find the prime factorization of $903$: $903 = 3 	imes 301 = 3 	imes 7 	imes 43$. The number of positive integral divisors of $903 = 3^1 	imes 7^1 	imes 43^1$ is $(1+1)(1+1)(1+1) = 2 	imes 2 	imes 2 = 8$.

If $[x]$ denotes the greatest integer function of $x$,then the number of positive integral divisors of $\left[(2+\sqrt{3})^5\right]$ is

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