Question

Larry has 4-cent stamps and 9-cent stamps, which he can combine to produce various amounts of postage. For example, he can make 40 cents by using four 9-cent stamps and a 4-cent stamp, or by using ten 4-cent stamps. However, there are some amounts of postage he can't make exactly, such as 10 cents. {nl} {nl} What is the largest number of cents that Larry cannot make exactly from a combination of 4- and/or 9-cent stamps? {nl} {nl} Explain how you know your answer is correct. (You should explain two things: why Larry can't make the amount of your answer, and why he can make any bigger amount.)

Answer 1

Answer:

• the largest number that cannot be made is 23
• see below for an explanation

Step-by-step explanation:

Why numbers greater than 23 can be made

From any number that Larry can make that includes an appropriate number of stamps, Larry can ...

• add 1 by adding 1×9 and taking away 2×4
• add 2 by adding 2×9 and taking away 4×4
• add 3 by adding 3×9 and taking away 6×4
• add 4 by adding 1×4
• add 5 by adding 4 and 1 (add 9, take away 4)
• add 6 by adding 4 and 2 (add 2×9, take away 3×4)
• add 7 by adding 4 and 3 (add 3×9, take away 5×4)
• add 8 by adding 2×4
• add 9 by adding 1×9

Clearly, for Larry to be able to continue adding any number, he must start with a number that contains at least 6×4 cents = 24 cents.

Larry can make 24 cents from 6 4-cent stamps. Using the above table, he can add any amount to that.

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Why 23 cannot be made

For 23 cents, the number of stamps must include 0, 1, or 2 9-cent stamps, so there are 3 cases to try for making 23:

• 23 cannot be made from 0 9-cent stamps because 23 is not divisible by 4
• 23 cannot be made using 1 9-cent stamp, because 14 is not divisible by 4
• 23 cannot be made using 2 9-cent stamps, because 5 is not divisible by 4.

Therefore, 23 cannot be made.