<s>[INST] 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 $5$
 [/INST] Note that
$$x \equiv 1 \mod 2 \\ x \equiv 2 \mod 3 \\ x \equiv 3 \mod 4 \\ x \equiv 4 \mod 5 \\ x \equiv 5 \mod 6$$
is equivalent to
$$x \equiv -1 \mod 2 \\ x \equiv -1 \mod 3 \\ x \equiv -1 \mod 4 \\ x \equiv -1 \mod 5 \\ x \equiv -1 \mod 6$$
And by the Chinese Remainder Theorem, the solution to the latter is easily found to be $x \equiv -1 \mod{\mathrm{lcm}(2,3,4,5,6)}$, or $x \equiv -1 \mod 60$.
So just count the number of $3$-digit numbers $x$ satisfying $x \equiv -1 \mod 60$. 
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