4.43. Find the smallest natural number that:

a) When divided by 5 gives a remainder of 4, when divided by 6 gives a remainder of 5, and when divided by 7 gives a remainder of 6.

b) When divided by 5 gives a remainder of 4, when divided by 6 gives a remainder of 5, and when divided by 8 gives a remainder of 7.
4.43. a) Answer: 209. Let $n$ be the desired number. Then $n+1$ is divisible by $5 \cdot 6 \cdot 7=210$. Therefore, $n=209$.

b) Answer: 119. Let $n$ be the desired number. Then $n+1$ is divisible by $\operatorname{LCM}(5,6,8)=120$. Therefore, $n=119$.