The number of distinct prime divisors of the | vedclass

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(C) Let $a = 512$,$b = -253$,and $c = -259$. Then $a + b + c = 512 - 253 - 259 = 0$. We know that if $a + b + c = 0$,then $a^3 + b^3 + c^3 = 3abc$. Th

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$6$

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(C) Let $a = 512$,$b = -253$,and $c = -259$. Then $a + b + c = 512 - 253 - 259 = 0$. We know that if $a + b + c = 0$,then $a^3 + b^3 + c^3 = 3abc$. Therefore,$(512)^3 + (-253)^3 + (-259)^3 = 3(512)(-253)(-259) = 3 	imes 512 	imes 253 	imes 259$. Now,factorize each term: $3 = 3^1$ $512 = 2^9$ $253 = 11 	imes 23$ $259 = 7 	imes 37$ So,the expression is $3^1 	imes 2^9 	imes 11^1 	imes 23^1 	imes 7^1 	imes 37^1$. The distinct prime divisors are $2, 3, 7, 11, 23, 37$. Thus,there are $6$ distinct prime divisors.

The number of distinct prime divisors of the number $(512)^3 - (253)^3 - (259)^3$ is

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