if x is a selected number of the number from 20−99, then what is probalility x^3 −x is divided by 12?


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Question Number 98773 by john santu last updated on 16/Jun/20

$if x is a selected number of the$ $number from 20 - 99, then what$ $is probalility x^{3} - x is divided by$ $12 ?$
	Answered by mr W last updated on 16/Jun/20

	$x^{3} - x = (x - 1) x (x + 1)$ $x - 1 = 12 k \Rightarrow x = 12 k + 1, k = 2 . . 8$ $x = 12 k \Rightarrow x = 12 k, k = 2 . . 8$ $x + 1 = 12 k \Rightarrow x = 12 k - 1, k = 2 . . 8$ $3 \times 7 = 21 p o s s i b l e v a l u e s f o r x$ $\Rightarrow p = \frac{21}{80}$
	Commented by mr W last updated on 16/Jun/20

	$y o u a r e r i g h t s i r!$ $p = \frac{60}{80} = \frac{3}{4}$
	Commented by mr W last updated on 16/Jun/20

	$c a s e 1 : x - 1 i s e v e n, i . e . x i s o d d,$ $i . e . x = 21, 23, . . ., 99 \Rightarrow 40 n u m b e r s$ $c a s e 2 : x i s d i v i s i b l e b y 4,$ $i . e . x = 20, 24, . . ., 96 \Rightarrow 20 n u m b e r s$ $\Rightarrow t o t a l l y 60 v a l u e s f o r x .$
	Commented by john santu last updated on 16/Jun/20

	$yes . thank you friend$
	Commented by MJS last updated on 16/Jun/20

	$x^{3} - x is always divisible by 3$ $thus we only need to check which numbers$ $are divisible by 4$
	Answered by MJS last updated on 16/Jun/20
	$n=x^3 −x=x(x−1)(x+1) ⇒ 3∣n ∀x∈Z ⇒ { ((x=2k+1 ⇒ 12∣n)),((x=2n ⇒ { ((x=4n ⇒ 12∣n)),((x=4n+2 ⇒ 12∤n)) :})) :} ⇒ for 20≤x≤99 we have 20 numbers for which 12∤n ⇒ searched probability is (3/4)$
	$n = x^{3} - x = x (x - 1) (x + 1)$ $\Rightarrow 3 ∣ n \forall x \in Z$ $\Rightarrow {\begin{cases} x = 2 k + 1 \Rightarrow 12 ∣ n \\ x = 2 n \Rightarrow {\begin{cases} x = 4 n \Rightarrow 12 ∣ n \\ x = 4 n + 2 \Rightarrow 12 ∤ n \end{cases} \end{cases}$ $\Rightarrow for 20 ⩽ x ⩽ 99 we have 20 numbers for$ $which 12 ∤ n$ $\Rightarrow searched probability is \frac{3}{4}$
	Commented by john santu last updated on 16/Jun/20

	$why the method mr W not same sir .$ $i got too \frac{21}{80}$
	Commented by MJS last updated on 16/Jun/20

	$n = x (x - 1) (x + 1)$ $x n$ $20 12 \times 665$ $21 12 \times 770$ $22 10626$ $23 12 \times 1012$ $24 12 \times 1150$ $25 12 \times 1300$ $26 17550$ $27 12 \times 1638$ $28 12 \times 1827$ $29 12 \times 2030$ $. . .$
	Commented by bemath last updated on 16/Jun/20

	$for x = 22, 26, . . . not satisfy sir$
	Commented by bemath last updated on 16/Jun/20

	$30 30 \times 29 \times 31 ∤ 12$ $34 34 \times 33 \times 35 ∤ 12$ $38 38 \times 37 \times 39 ∤ 12$ $correct sir ?$
	Commented by MJS last updated on 16/Jun/20

	$y e s$
	Commented by john santu last updated on 16/Jun/20

	$thank you sir$
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