You can clean 21 windowed office building on a certain number of buildings. They can also clean the windows of a school with 42 windows. Altogether there are enough supplies to clean 126 windows. How many office buildings can be washed

Let \( x \) be the number of windowed office buildings cleaned, and each office building has 21 windows. Let \( y \) be the number of schools cleaned, and the school has 42 windows.

According to the problem, the total number of windows cleaned is given by:

\[
21x + 42y = 126
\]

To simplify this equation, we can divide everything by 21:

\[
x + 2y = 6
\]

Now, we need to find non-negative integer solutions for \( x \) and \( y \).

We can rewrite the equation to express \( x \):

\[
x = 6 - 2y
\]

Now we will determine the possible values for \( y \) that keep \( x \) non-negative:

1. If \( y = 0 \):
\[
x = 6 - 2(0) = 6
\]
Therefore, \( (x, y) = (6, 0) \).

2. If \( y = 1 \):
\[
x = 6 - 2(1) = 4
\]
Therefore, \( (x, y) = (4, 1) \).

3. If \( y = 2 \):
\[
x = 6 - 2(2) = 2
\]
Therefore, \( (x, y) = (2, 2) \).

4. If \( y = 3 \):
\[
x = 6 - 2(3) = 0
\]
Therefore, \( (x, y) = (0, 3) \).

If \( y \) were to take a value of 4 or more, then \( x \) would become negative, which is not allowed.

Thus, the possible non-negative integer solutions for \( (x, y) \) are:

- \( (6, 0) \): Clean 6 office buildings, 0 schools.
- \( (4, 1) \): Clean 4 office buildings, 1 school.
- \( (2, 2) \): Clean 2 office buildings and 2 schools.
- \( (0, 3) \): Clean 0 office buildings and 3 schools.

Therefore, the maximum number of office buildings that can be washed is:

\[
\boxed{6}
\]