The sum $1+1+4$ of the digits of the number 114 divides the number itself. What is the largest number, less than 900, that satisfies this property?
To find the largest possible number less than 900, we start with the digit 8 in the hundreds place. We observe that the number 800 satisfies the property. Therefore, the number we are looking for is greater than or equal to 800.

We need to find $a$ and $b$ such that $8+a+b$ divides $8ab = 800 + 10a + b$. We remember that $8+a+b$ divides $8ab = 800 + 10a + b$ if and only if $8+a+b$ divides $800 + 10a + b - (8 + a + b) = 792 + 9a$. Now, assigning values to $a$ in descending order, we get:

- $a=9 \Rightarrow 792 + 9 \times 9 = 873 = 9 \times 97$ and this number has no divisor between $17 (b=0)$ and $26 (b=9)$.
- $a=8 \Rightarrow 792 + 9 \times 8 = 864 = 2^5 \times 3^3$. The largest divisor of this number between $16$ and $25$ is $24$, that is, $b=8$. Therefore, the number we are looking for is 888.