13. The wealthy Croesus buys 88 identical vases. The price of each, expressed in drachmas, is an integer (the same for all 88 vases). We know that Croesus pays a total of $a 1211 b$ drachmas, where $a, b$ are digits to be determined (which may be distinct or the same). How many drachmas does a single vase cost?
13. The answer is 1274. We notice that the number $a 1211 b$ must be divisible by 88, so it must be divisible by both 11 and 8. We know that a number is divisible by 8 if and only if its last three digits are, so $11 b$ must be divisible by 8. We observe that $8 \cdot 14=112$ is the only number of the form $11 b$ divisible by 8, so $b=2$ and $a 12112$ is divisible by 8 for any choice of $a$. The divisibility rule for 11 tells us that the difference between the sum of the digits in even positions and the sum of the digits in odd positions must be a multiple of 11, meaning that $(2+1+1)-(1+2+a)=1-a$ must be divisible by 11. Since $a$ is a digit, thus between 0 and 9, this condition is only satisfied for $a=1$. The solution is therefore $\frac{112112}{88}=1274$.