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 $5$

Observe that $119$ leaves remainder $n-1$ when divided by $n$  where $2\le n\le 6$
So, $119+1$ is divisible by $n$ for $2\le n\le 6$
So, we numbers of the form $\operatorname{lcm}(2,3,4,5,6)m-1=60m-1$ where $m$ is any positive integer. 
As the required number is of three digits $100\le 60m-1\le 999\implies 2\le m\le 16$ 
Hence there are $16-2+1=15$ such numbers.
