What is the largest amount of postage in cents that cannot be made using only 3 cent and 5 cent stamps?
(A) 7
(B) 13
(C) 4
(D) 8
(E) 9
Solution 1

We look at each of the choices and try to make them using only 3 cent and 5 cent stamps:

(A): 7 cannot be made, since no more than one 5 cent and two 3 cent stamps could be used (try playing with the possibilities!)

(B): $13=5+5+3$

(C): 4 cannot be the answer since a larger number (7) already cannot be made

(D): $8=5+3$

(E): $9=3+3+3$

Therefore, the answer must be 7 .

(We have not really justified that 7 is the largest number that cannot be made using only $3 \mathrm{~s}$ and 5 s; we have, though, determined that 7 must be the answer to this question, since it is the only possible answer from the given possibilities! See Solution 2 for a justification that 7 is indeed the answer.)

## Solution 2

We make a table to determine which small positive integers can be made using $3 \mathrm{~s}$ and $5 \mathrm{~s}$ :

| Integer | Combination |
| :---: | :---: |
| 1 | Cannot be made |
| 2 | Cannot be made |
| 3 | 3 |
| 4 | Cannot be made |
| 5 | 5 |
| 6 | $3+3$ |
| 7 | Cannot be made |
| 8 | $5+3$ |
| 9 | $3+3+3$ |
| 10 | $5+5$ |
| 11 | $5+3+3$ |

Every integer larger than 11 can also be made because the last three integers in our table can be made and we can add a 3 to our combinations for 9,10 and 11 to get combinations for 12, 13 and 14 , and so on.

From the table, the largest amount of postage that cannot be made is 7 .

ANSWER: (A)