Question

Only genious (not the rank) can do it :p
Did you find ?
So answer :)
Thanks.

Answer 1

$\bold{\large{\underline{\underline{Answer}}}}$

Consider the number be x

when,

$\bold{ \dfrac{x}{2}\: , \: R \: = \: 1}$

$\bold{ \dfrac{x}{3}\: , \: R \: = \: 2}$

$\bold{ \dfrac{x}{4}\: , \: R \: = \: 3}$

and so on to $\bold{ \dfrac{x}{10}\: , \: R \: = \: 9}$

Now , go to back studies we have a statement

If the smallest number which when divided by p,q and r leaves remainder x,y and z where (p-x) = ( q-y) = (r-z) = constant then the number will be LCM(p,q,r) - constant

here ,

2-1 = 1 , 3-2 = 1 , 4-3 = 1 to 10-9 = 1

We get $\bold{1}$ as a constant.

So

$\bold{x\: =\: LCM(2,3,4,5,6,7,8,9,10)\:-\:1}$

$\begin{center} \begin{tabular}{|c|c} 2 & 2,3,4,5,6,7,8,9,10\\ \cline{1-2} 2 & 1,3,2,5,3,7,4,9,5\\ \cline{1-2} 2 & 1,3,1,5,3,7,2,9,5\\ \cline{1-2} 3 & 1,3,1,5,3,7,1,9,5\\ \cline{1-2} 3 & 1,1,1,5,1,7,1,3,5\\ \cline{1-2} 5 & 1,1,1,5,1,7,1,1,5\\ \cline{1-2} 7 & 1,1,1,1,1,7,1,1,5\\ \cline{1-2} & 1,1,1,1,1,1,1,1,1\\ \end{tabular} \end{center}$

x = (2×2×2×3×3×5×7)-1

x = 2,520 -1

x = 2,519

The number is $\bold{2,519}$

$\bold{\large{Thanks}}$

Answer 2

● Hope this answer will help u 
N + 1 is divisible by 2, 3, 4, 5, 6, 7, 8, 9 & 10 so clearly one solution would be 10! (ie 10*9*8*7*6*5*4*3*2 = 3628800 for N +1) but this is not the lowest possible. Follow the logic below... 

N + 1 must be a multiple of 2 

N + 1 must be a multiple of 4 but if it is a multiple of 2 & 4 it is necessarily a multiple of 8 

N + 1 must be a multiple of 5 but if it is a multiple of 2 & 5 it is necessarily a multiple of 10 

N + 1 must be a multiple of 6 but if it is a multiple of 2 & 9 (18) it is necessarily a multiple of 6 

N + 1 must be a multiple of 7 

N + 1 must be a multiple of 9 

Hence 2 x 4 x 5 x 7 x 9 =$\color{red}\mathcal{2520}$
2520  is a multiple of 2, 3, 4, 5, 6, 7, 8, 9 & 10 
 
$\color{green}\huge\bold\star\underline\mathcal{neymarjr3}\star$