# Problem of the Week Problem C and Solution Wipe Away 1

## Problem

Chetan writes the positive integers from $$1$$ to $$200$$ on a whiteboard. Wassim then erases all the numbers that are multiples of $$9$$. Karla then erases all the remaining numbers that contain the digit $$9$$. How many numbers are left on the whiteboard?

## Solution

We first calculate the number of integers that Wassim erases, which is the number of multiples of $$9$$ between $$1$$ and $$200$$. Since $$200 = (22 \times 9) + 2$$, there are $$22$$ multiples of $$9$$ between $$1$$ and $$200$$. Thus, Wassim erases $$22$$ numbers from the whiteboard.

Now letâ€™s figure out how many of the integers from $$1$$ to $$200$$ contain the digit $$9$$. From $$1$$ to $$100$$, these numbers are $$9, 19, \ldots, 79, 89$$ as well as $$90, 91, \ldots , 98, 99$$. There are $$19$$ of these numbers from $$1$$ to $$100$$. There are another $$19$$ between $$101$$ and $$200$$, which are obtained by adding $$100$$ to each of the numbers from $$1$$ to $$100$$. Therefore, $$19 + 19 =38$$ integers from $$1$$ to $$200$$ contain the digit $$9$$.

However, some of the integers that contain the digit $$9$$ are also multiples of $$9$$, so were erased by Wassim. There are $$5$$ of these numbers between $$1$$ and $$200$$: $$9$$, $$90$$, $$99$$, $$189$$, and $$198$$. Thus, Karla erases $$38-5=33$$ numbers from the whiteboard.

Hence, the number of numbers left on the whiteboard is $$200 - 22 - 33 = 145$$.