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.)

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.