**ANSWER**:

*43.*
**EXPLANATION**: The different number of doughnuts that can be purchased can be found by
plugging in all the different combinations of values for the number of 6, 9, and 20 doughnut boxes in the
following equation:

d = 6x + 9y + 20z

To do that, make a list of all the multiples of 6, 9, and 20.

Multiples of

**6**: 6, 12, 18, 24, 30, 36, 42, 48...

Multiples of

**9**: 9, 18, 27, 36, 45...

Multiples of

**20**: 20, 40...

If you look at the lists of the multiples of 6 and 9, you'll notice that between the two lists and adding
a number from each list together, you can make all the multiples of 3 beginning with 6. For example, you
can purchase 15 doughnuts by adding 6 (one box of 6 doughnuts) and 9 (one box of 9 doughnuts). So the list
of multiples of 6 and 9 can be combined into one list containing the multiples of 3 beginning with 6:

Multiples of

**3** (beginning with 6): 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48...

Now add 20 and 40 to each number on the list (for the boxes of 20 doughnuts) and include any new totals:

6, 9, 12, 15, 18, 20, 21, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49...

Once you can make 6 consecutive numbers (in this case, 44, 45, 46, 47, 48, 49), you know that you can make
any other number simply by adding a box of six doughnuts. So the largest number of doughnuts that CANNOT
be purchased is 43.