how many number like $119$ How many 3-digits number has this property like $119$:
$119$ divided by $2$ the remainder is $1$ 
119 divided by $3$ the remainderis $2 $
$119$ divided by $4$ the remainder is $3$
$119$ divided by $5$ the remainder is $4$
$119$ divided by $6$ the remainder is $5$

Exercises: 
Try dividing $420 - 1$ by integers $2, 3, ..., 7$ and note the respective remainders are $1, 2, 3, 4, 5, 6$. 
Then note that $420 = \textrm{lcm}\,(2, 3, 4, 5, 6, 7)$
Any $n = k\cdot 420 - 1$, $k\in \mathbb{Z}$, will yield the same remainders when divided by $2, 3, ..., 7$.

Try dividing $840 - 1 = 839\,$ by $\,2,\, 3, \,4, \,5, \,6, \,7,\,8\,$ $\implies$ respective remainders of $\;$...?...
Then note that $840 = \textrm{lcm}\,(2, 3, 4, 5, 6, 7, 8)$.
So the same respective remainders are obtained, when $k\cdot 840 - 1$ is divided by each of $\{2, 3, ..., 8\}$. 

Now: What\s the least positive number, that when divided by each of $\{2, 3, 4, 5, 6, 7, 8, 9\}$ gives a corresponding remainder of $\{1, 2, 3, 4, 5, 6, 7, 8\}$? 
Find the $\text{lcm}\,(2, 3, 4, 5, 6, 7, 8, 9\} = 2^3\cdot 3^2\cdot 5 \cdot 7 = 840\cdot 3 = 2520.\;$ 
Subtract $1:\;\;$ $2520 - 1 = 2519 = n_9$.
Test by dividing $n_9$ by each of $2, 3, 4, 5, 6, 7, 8, 9$, to confirm...
There you go.
