if-x-is-a-selected-number-of-the-number-from-20-99-then-what-is-probalility-x-3-x-is-divided-by-12-

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, \dots, 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, \dots, 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)

\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

\dots

Commented by bemath last updated on 16/Jun/20

for x = 22, 26, \dots 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