11-4. At first, a natural number was displayed on the calculator screen. Each time, Tanya added to the current number \( n \) on the screen a natural number that \( n \) did not divide. For example, if the screen showed the number 10, Tanya could add 7 to get 17.

Tanya repeated this operation five times, and the number 100 appeared on the screen. What is the largest initial number for which this could have happened?
Answer: 89.

Solution. Estimation. Note that Tanya increased the number on the screen by at least 2 each time, because any number is divisible by 1. If Tanya added two five times, the initial number would have been 90, and it would not have been possible to add two to it. Therefore, Tanya must have added a number greater than two at least once. Consequently, she increased the number by at least 11 in total, from which $n \leqslant 89$.

Estimation. For $n=89$, this is possible: $100=89+2+2+2+2+3$.