3. For tasks a) and b), given the numbers $A$ and $B$, determine whether there exists a natural number that, when divided by $A$, gives a remainder of 1, and when divided by $B$, gives a remainder of 2: a) $A=6, B=8$; b) $A=7, B=9$.
Solution. a) If such a natural number $C$ exists, then on the one hand, when divided by 6 it gives a remainder of 1, which means $C$ is an odd number, and on the other hand, when divided by 8 it gives a remainder of 2, which means $C$ is an even number. This contradiction implies that such a natural number $C$ does not exist. b) For example, the number 29, when divided by 7 and 9, gives remainders of 1 and 2, respectively.

Answer: a) no; b) yes.

Comment. Full solution a) - 5 points, b) - 2 points.