My calculator has one of its keys broken so that when pressed it appears a number that is not the one corresponding to the key. If I type all the numbers from 9 to 1: 987654321, a number divisible by 11 appears on the screen and leaves 3 remainder when divided by 9.

**What is the broken key?**

#### Solution

The number 987654321 is a multiple of nine since their numbers add up to 45. To give the remainder 3, we must increase the sum by 3 units or decrease it by 6 units.

On the other hand, the sum of the numbers of the odd places is 25 and the sum of the figures of the even places is 20. The difference between the two, 5, must have a lag of 6 more units to be a multiple of 11.

Decreasing by 6 units the sum of the numbers of the even places we will obtain a multiple of 11 and also dividing it by 9 will give us the rest 3.

This can only be achieved if we change the number 8 to 2. Therefore **the broken key corresponds to the number 8** and the number that appears on the calculator is 927654321.