4.44. Find the number that:

a) When divided by 5 gives a remainder of $a$, and when divided by $6$ gives a remainder of $b$.

b) When divided by 5 gives a remainder of $a$, when divided by $6$ gives a remainder of $b$, and when divided by 7 gives a remainder of $c$.
4.44. a) Answer: $6 a+25 b$. Let $6 n \equiv 1(\bmod 5)$ and $5 m \equiv 1(\bmod 6)$. Then

$$
\begin{aligned}
& 6 n a+5 m b \equiv 6 n a \equiv a(\bmod 5) \\
& 6 n a+5 m b \equiv 5 m b \equiv b(\bmod 6) .
\end{aligned}
$$

In our case $m=5$ and $n=1$.

b) Answer: $126 a+175 b+120 c$. Let $42 k \equiv 1(\bmod 5), 35 n \equiv$ $\equiv 1(\bmod 6)$ and $30 m \equiv 1(\bmod 7)$ (here $42=6 \cdot 7,35=5 \cdot 7,30=5 \cdot 6)$. Then

$$
\begin{aligned}
& 42 k a+35 n b+30 m c \equiv 42 k a \equiv a(\bmod 5) \\
& 42 k a+35 n b+30 m c \equiv 35 n b \equiv b(\bmod 6) \\
& 42 k a+35 n b+30 m c \equiv 30 m c \equiv c(\bmod 7)
\end{aligned}
$$

In our case $k=3, n=5, m=4$.