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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?

$$\mathrm{if}\:\mathrm{x}\:\mathrm{is}\:\mathrm{a}\:\mathrm{selected}\:\mathrm{number}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{from}\:\mathrm{20}−\mathrm{99},\:\mathrm{then}\:\mathrm{what} \\ $$$$\mathrm{is}\:\mathrm{probalility}\:\mathrm{x}^{\mathrm{3}} −\mathrm{x}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\mathrm{12}?\: \\ $$

Answered by mr W last updated on 16/Jun/20

x^3 −x=(x−1)x(x+1)  x−1=12k ⇒x=12k+1, k=2..8  x=12k ⇒x=12k, k=2..8  x+1=12k ⇒x=12k−1, k=2..8  3×7=21 possible values for x  ⇒p=((21)/(80))

$${x}^{\mathrm{3}} −{x}=\left({x}−\mathrm{1}\right){x}\left({x}+\mathrm{1}\right) \\ $$$${x}−\mathrm{1}=\mathrm{12}{k}\:\Rightarrow{x}=\mathrm{12}{k}+\mathrm{1},\:{k}=\mathrm{2}..\mathrm{8} \\ $$$${x}=\mathrm{12}{k}\:\Rightarrow{x}=\mathrm{12}{k},\:{k}=\mathrm{2}..\mathrm{8} \\ $$$${x}+\mathrm{1}=\mathrm{12}{k}\:\Rightarrow{x}=\mathrm{12}{k}−\mathrm{1},\:{k}=\mathrm{2}..\mathrm{8} \\ $$$$\mathrm{3}×\mathrm{7}=\mathrm{21}\:{possible}\:{values}\:{for}\:{x} \\ $$$$\Rightarrow{p}=\frac{\mathrm{21}}{\mathrm{80}} \\ $$

Commented by mr W last updated on 16/Jun/20

you are right sir!  p=((60)/(80))=(3/4)

$${you}\:{are}\:{right}\:{sir}! \\ $$$${p}=\frac{\mathrm{60}}{\mathrm{80}}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Commented by mr W last updated on 16/Jun/20

case 1: x−1 is even, i.e. x is odd,  i.e. x=21, 23, ..., 99 ⇒ 40 numbers  case 2: x is divisible by 4,  i.e. x=20, 24, ..., 96 ⇒ 20 numbers  ⇒totally 60 values for x.

$${case}\:\mathrm{1}:\:{x}−\mathrm{1}\:{is}\:{even},\:{i}.{e}.\:{x}\:{is}\:{odd}, \\ $$$${i}.{e}.\:{x}=\mathrm{21},\:\mathrm{23},\:...,\:\mathrm{99}\:\Rightarrow\:\mathrm{40}\:{numbers} \\ $$$${case}\:\mathrm{2}:\:{x}\:{is}\:{divisible}\:{by}\:\mathrm{4}, \\ $$$${i}.{e}.\:{x}=\mathrm{20},\:\mathrm{24},\:...,\:\mathrm{96}\:\Rightarrow\:\mathrm{20}\:{numbers} \\ $$$$\Rightarrow{totally}\:\mathrm{60}\:{values}\:{for}\:{x}. \\ $$

Commented by john santu last updated on 16/Jun/20

yes. thank you friend

$$\mathrm{yes}.\:\mathrm{thank}\:\mathrm{you}\:\mathrm{friend}\: \\ $$

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

$${x}^{\mathrm{3}} −{x}\:\mathrm{is}\:\mathrm{always}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{3} \\ $$$$\mathrm{thus}\:\mathrm{we}\:\mathrm{only}\:\mathrm{need}\:\mathrm{to}\:\mathrm{check}\:\mathrm{which}\:\mathrm{numbers} \\ $$$$\mathrm{are}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{4} \\ $$

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)

$${n}={x}^{\mathrm{3}} −{x}={x}\left({x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right) \\ $$$$\Rightarrow\:\mathrm{3}\mid{n}\:\forall{x}\in\mathbb{Z} \\ $$$$\Rightarrow\:\begin{cases}{{x}=\mathrm{2}{k}+\mathrm{1}\:\Rightarrow\:\mathrm{12}\mid{n}}\\{{x}=\mathrm{2}{n}\:\Rightarrow\:\begin{cases}{{x}=\mathrm{4}{n}\:\Rightarrow\:\mathrm{12}\mid{n}}\\{{x}=\mathrm{4}{n}+\mathrm{2}\:\Rightarrow\:\mathrm{12}\nmid{n}}\end{cases}}\end{cases} \\ $$$$\Rightarrow\:\mathrm{for}\:\mathrm{20}\leqslant{x}\leqslant\mathrm{99}\:\mathrm{we}\:\mathrm{have}\:\mathrm{20}\:\mathrm{numbers}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{12}\nmid{n} \\ $$$$\Rightarrow\:\mathrm{searched}\:\mathrm{probability}\:\mathrm{is}\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Commented by john santu last updated on 16/Jun/20

why the method mr W not same sir.  i got too((21)/(80))

$$\mathrm{why}\:\mathrm{the}\:\mathrm{method}\:\mathrm{mr}\:\mathrm{W}\:\mathrm{not}\:\mathrm{same}\:\mathrm{sir}. \\ $$$$\mathrm{i}\:\mathrm{got}\:\mathrm{too}\frac{\mathrm{21}}{\mathrm{80}} \\ $$

Commented by MJS last updated on 16/Jun/20

n=x(x−1)(x+1)  x     n  20   12×665  21   12×770  22   10626  23   12×1012  24   12×1150  25   12×1300  26   17550  27   12×1638  28   12×1827  29   12×2030  ...

$${n}={x}\left({x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right) \\ $$$${x}\:\:\:\:\:{n} \\ $$$$\mathrm{20}\:\:\:\mathrm{12}×\mathrm{665} \\ $$$$\mathrm{21}\:\:\:\mathrm{12}×\mathrm{770} \\ $$$$\mathrm{22}\:\:\:\mathrm{10626} \\ $$$$\mathrm{23}\:\:\:\mathrm{12}×\mathrm{1012} \\ $$$$\mathrm{24}\:\:\:\mathrm{12}×\mathrm{1150} \\ $$$$\mathrm{25}\:\:\:\mathrm{12}×\mathrm{1300} \\ $$$$\mathrm{26}\:\:\:\mathrm{17550} \\ $$$$\mathrm{27}\:\:\:\mathrm{12}×\mathrm{1638} \\ $$$$\mathrm{28}\:\:\:\mathrm{12}×\mathrm{1827} \\ $$$$\mathrm{29}\:\:\:\mathrm{12}×\mathrm{2030} \\ $$$$... \\ $$

Commented by bemath last updated on 16/Jun/20

for x =22 , 26 , ... not satisfy sir

$$\mathrm{for}\:\mathrm{x}\:=\mathrm{22}\:,\:\mathrm{26}\:,\:...\:\mathrm{not}\:\mathrm{satisfy}\:\mathrm{sir} \\ $$

Commented by bemath last updated on 16/Jun/20

30   30×29×31 ∤ 12  34    34×33×35 ∤12  38    38×37×39 ∤ 12  correct sir?

$$\mathrm{30}\:\:\:\mathrm{30}×\mathrm{29}×\mathrm{31}\:\nmid\:\mathrm{12} \\ $$$$\mathrm{34}\:\:\:\:\mathrm{34}×\mathrm{33}×\mathrm{35}\:\nmid\mathrm{12} \\ $$$$\mathrm{38}\:\:\:\:\mathrm{38}×\mathrm{37}×\mathrm{39}\:\nmid\:\mathrm{12} \\ $$$$\mathrm{correct}\:\mathrm{sir}? \\ $$

Commented by MJS last updated on 16/Jun/20

yes

$${yes} \\ $$

Commented by john santu last updated on 16/Jun/20

thank you sir

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$

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