jmc

First, we look for an integer which leaves remainder of 1 when divided by 4 and a remainder of 2 when divided by 5. Checking the remainders of 2, 7, 12, 17, $\dots$ when divided by 4, we find that 17 is the least positive integer satisfying this condition. By the Chinese Remainder Theorem, the only positive integers which leave a remainder of 1 when divided by 4 and a remainder of 2 when divided by 5 are those that differ from 17 by a multiple of $4\cdot 5=20$. Checking the remainders of 17, 37, $\dots$ when divided by 7, we find that $17$ leaves a remainder of 3. Again, using the Chinese Remainder Theorem, the integers which satisfy all three conditions are those that differ from 17 by a multiple of $4\cdot 5\cdot 7=140$. Among the integers 17, 157, 297, $\dots$, only $\boxed{157}$ is between 100 and 200.