In the number $6 a 78 b$, $a$ represents the thousands digit and $b$ represents the units digit. If $6 a 78 b$ is divisible by 45, then the value of $a+b$ is:
(a) 5 ;
(b) 6 ;
(c) 7 ;
(d) 8 ;
(e) 9 .
The correct option is (b).

The number is divisible by $45=5 \times 9$, so it is divisible by 5 and 9. Every number divisible by 5 ends in 0 or 5. Thus, $b=0$ or $b=5$. Every number divisible by 9 has the sum of its digits as a multiple of 9. Therefore, $6+a+7+8+b=21+a+b$ is a multiple of 9. Since $a \leq 9$, and $b=0$ or $b=5$, we have $21 \leq 21+a+b \leq 21+9+5=35$. But the only multiple of 9 between 21 and 35 is 27. Thus, $21+a+b=27$. We conclude that $a+b=6$ and the number sought is 61785 or 66780.