The number 119 is very interesting because it leaves a remainder of 1 when divided by 2, a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, a remainder of 4 when divided by 5, and finally, a remainder of 5 when divided by 6. Are there other three-digit numbers with these properties?
Suppose $N$ is one of the numbers sought. Since $N$ and 119 leave the same remainders when divided by 2, 3, 4, 5, and 6, the difference $N-119$ between them leaves a remainder of zero when divided by these numbers. Therefore, $N-119$ is a multiple of 2, 3, 4, 5, and 6. Since 60 is the least common multiple of these numbers, $N-119$ is a multiple of 60, that is, $N-119=60k$, for some integer $k$. Thus, $N=119+60k$ is an interesting number for any integer $k$. As we want numbers distinct from 119 and with three digits, we must take $k$ positive and less than 15, since $119+60 \times 15=1019$ has four digits. Thus, we take $k$ from 1 to 14 and obtain 14 other interesting three-digit numbers, namely,

179, 239, 299, 359, 419, 479, 539, 599, 659, 719, 779, 839, 899, 959.