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12 January, 00:59

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.)

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  1. 12 January, 01:08
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    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.

    ___

    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.
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